OCEAN THERMAL ENERGY CONVERSION PLANTS:
EXPERIMENTAL AND ANALYTICAL STUDY
OF MIXING AND RECIRCULATION
by
Gerhard H. Jirka
David J. Fry
Energy Laboratory
Report No. MIT-EL 77-011
R. Peter Johnson
Donald R.F. Harleman
September 1977
______
OCEAN THERMAL ENERGY CONVERSION PLANTS:
EXPERIMENTAL AND ANALYTICAL STUDY OF MIXING AND RECIRCULATION
by
Gerhard H. Jirka
R. Peter Johnson
David J. Fry
Donald R.F. Harleman
ENERGY LABORATORY
in association with
RALPH M. PARSONS LABORATORY FOR
WATER RESOURCES AND HYDRODYNAMICS
DEPARTMENT OF CIVIL ENGINEERING
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Energy Laboratory Report No. MIT-EL 77-011
September 1977
Prepared under the support of
Division of Solar Energy
U.S. Energy Research and Development Administration
Washington, D.C.
Contract No. EY-76-S-02-2909.MO01
ABSTRACT
Ocean thermal energy conversion (OTEC) is a method of generating
power using the vertical temperature gradient of the tropical ocean as an
energy source.
Experimental and analytical studies have been carried out to
determine the characteristics of the temperature and velocity fields induced
in the surrounding ocean by the operation of an OTEC plant.
The condition of
recirculation, i.e. the re-entering of mixed discharge water back into the
plant intake, was of particular interest because of its adverse effect on plant
The studies were directed at the mixed discharge concept, in which
efficiency.
the evaporator and condenser water flows are exhausted jointly at the approximate level of the ambient ocean thermocline.
The OTEC plant was of the
A
symmetric spar-buoy type with radial or separate discharge configurations.
distinctly stratified ocean with uniform, ambient current velocity was assumed.
The following conclusions are obtained:
The recirculation potential of an OTEC plant in a stagnant ocean is
determined by the interaction of the jet discharge zone and a double sink
return flow (one sink being the evaporator intake, the other the jet entrainment).
This process occurs in the near-field of an OTEC plant up to a
distance of about three times the ocean mixed layer depth.
The stratified
internal flow beyond this zone has little effect on recirculation, as have
small ocean current velocities (up to 0.10 m/s prototype).
Conditions which
are conducive to recirculation are characterized by high discharge velocities
and large plant flow rates.
A design formula is proposed which determines
whether recirculation would occur or not as a function of plant design and ocean
conditions.
On the basis of these results, it can be concluded that a 100 MW
OTEC plant with the mixed discharge mode can operate at a typical candidate
ocean site without incurring any discharge recirculation.
ACKNOWLEDGEMENTS
This study was sponsored by the Division of Solar Energy, U.S. Energy
Research and Development Administration, under contract No. E(11-1)-2909 with
the M.I.T. Energy Laboratory and the R.M. Parsons Laboratory and was adminThe
istered by the M.I.T. Office of Sponsored Programs, Account No. 83746.
ERDA program officers were Dr. Robert Cohen during the initial stage and
Dr. Lloyd Lewis for the remainder of the project.
Program support was also
provided by Dr. Jack Ditmars and his staff at the Argonne National Laboratory.
The cooperation and assistance of these individuals is gratefully acknowledged.
The experimental work was conducted at the R.M. Parsons Laboratory.
The assistance of Messrs. Edward F. McCaffrey, Electronics Engineer, Roy G.
Milley, Arthur Rudolph, Machinists, and Jeffrey Freudberg, Undergraduate Student,
was invaluable for completion of the project.
the M.I.T. Information Processing Center.
Computer work was performed at
The report was aptly typed by
Ms. Carole Bowman and Ms. Anne Clee.
The experimental and numerical studies were conducted by Mr. R. Peter
Johnson and Mr. David J. Fry, Graduate Research Assistants.
Research guidance
and supervision was given by Dr. Gerhard H. Jirka, Research Engineer in the
M.I.T. Energy Laboratory, and Dr. Donald R. F. Harleman, Ford Professor of
Engineering and Director of the R.M. Parsons Laboratory.
2
TABLE OF CONTENTS
Page
ABSTRACT
1
ACKNOWLEDGEMENTS
2
TABLE OF CONTENTS
3
CHAPTER
I
INTRODUCTION
7
1.1
1.2
Principles of OTEC Operation
Current Prototype Designs and the Need
7
for this Investigation
9
1.3
CHAPTER II
Purpose of the Study
PROTOTYPE SCHEMATIZATION AND SCALE MODELING
PARAMETERS
2.1
2.2
Physical Characteristics of the
Prototype
Schematization
2.2.1
2.2.2
2.2.3
2.3
CHAPTER III
11
11
11
14
14
18
21
EXPERIMENTAL LAYOUT AND OPERATION
30
3.1
3.2
3.3
The Basin
The OTEC Model
The Discharge and Intake Water
30
30
Circuits
34
3.4
3.5
3.6
3.7
3.8
3.9
CHAPTER IV
Schematization of the Ocean
Schematization of the Plant
Further Simplification for
Experimental Purposes
Model Scaling Parameters
10
Temperature Data Acquisition,
Processing and Presentation
Steady State Determination
Direct Recirculation Measurements with
Fluorescent Tracer
Fast Response Temperature Measurements
Velocity Measurements
Experimental Procedure
EXPERIMENTAL RESULTS:
STAGNANT OCEAN
4.1
4.2
36
39
40
40
41
42
RADIAL DISCHARGE AND
44
Run Conditions
Discussion of Results
44
47
4.2.1
4.2.2
47
Temperature Field Measurements
Steady-State Determination
Results
3
50
Page
4.2.3
4.2.4
4.2.5
4.3
4.4
CHAPTER V
Experimental Correlation of Discharge
Behavior
59
Extreme Case Experiments
63
65
5.1
5.2
Run Conditions
Discussion of Results
65
67
5.2.1
5.2.2
5.2.3
69
72
74
Temperature Field Measurements
Steady-State Determination
Recirculation Indications
Experimental Correlation of Discharge
Behavior
76
5.3.1
5.3.2
76
Jet Behavior
Characteristics of the Lateral
Return Flow
EXPERIMENTAL RESULTS:
81
OCEAN CURRENT
CONDITIONS
85
6.1
6.2
Run Conditions
Experimental Results
85
87
6.2.1
90
6.3
CHAPTER VII
54
54
57
EXPERIMENTAL RESULTS: SEPARATE JET DISCHARGE AND STAGNANT OCEAN
5.3
CHAPTER VI
Velocity Measurements
Fast Probe Measurements
Recirculation Measurements
Temperature Field Measurements
Experimental Correlations of Discharge Behavior
93
RADIAL JET THEORY WITH STAGNANT CONDITIONS
98
7.1
Method
98
7.2
Model for Radial Buoyant Jet Mixing
with Return Flow
101
7.2.1
7.2.2
102
107
7.3
7.4
of Analysis
Zone of Established Flow
Zone of Flow Establishment
Solution Method
Model Results
7.4.1
110
111
Surface Jet (Laboratory Model)
Predictions
7.4.1.1
7.4.1.2
4
Discharge Parameter
Sensitivity
Model Coefficient
Sensitivity
112
112
113
Page
7.4.2
7.5
Interface Jet (Prototype)
Comparison of Analytical and Experimental Results
Low Froude Number Experiments
(without recirculation)
7.5.2 High Froude Number Experiments
(with recirculation)
118
123
7.5.1
CHAPTER VIII
134
SEPARATE JET THEORY WITH STAGNANT CONDITIONS
142
8.1
Analytical Background
142
8.1.1
8.1.2
8.1.3
143
148
148
8.2
8.3
8.4
Zone of Established Flow
Zone of Flow Establishment
Interface Jet
Solution Method
Model Results
150
150
8.3.1
8.3.2
150
155
Surface Jet
Interface Jet (Prototype)
Comparison of Analytical and Experimental Results
8.4.1
8.4.2
8.4.3
CHAPTER IX
124
Low Froude Number Experiments
High Froude Number Experiments
Discharge Froude Number
Sensitivity
155
159
165
168
CONCLUSIONS AND DESIGN CONSIDERATIONS
173
9.1
9.2
9.3
173
174
177
Summary
Conclusions
Design Considerations
9.3.1
9.3.2
9.3.3
Mixed versus Non-Mixed Discharges
Design Formula
Intake Design
REFERENCES
178
181
185
186
APPENDIX A
Experimental Isotherm Data
A.1
A.2
A.3
Radial Stagnant Experiments
Separate Jet Stagnant Experiments
Experiments with an Ambient Current
188
189
199
218
APPENDIX B
Velocity Measurement Through Dye Photographs
248
APPENDIX C
Fast Probe Temperature Measurements
252
5
Page
APPENDIX D
Development
of the Equations
for the Radial
Jet Discharge
APPENDIX E
Development
257
of the Pressure
Derivative
in the
Vertically Integrated Momentum Equation
6
266
CHAPTER
I
INTRODUCTION
1.1
Principles of OTEC Operation
Ocean thermal energy conversion (OTEC) is a method of generating
power using the vertical temperature gradient of the tropical ocean as
The upper layer of the ocean collects energy by
an energy source.
The underlying water is colder due
changing solar radiation to heat.
to the return flow from polar regions which occurs within the global
circulation
of the world ocean.
Ocean thermal energy conversion to produce economically usable
forms of power is based on the same thermodynamic principles employed
in conventional methods of power production.
The thermal difference
between the warm upper water and the cold lower water is used to
vaporize and condense, respectively, a working fluid which, in turn,
An OTEC plant differs from conventional power plants
drives turbines.
in that is has a very low thermodynamic efficiency.
Efficiency, e, based on the second law of thermodynamics, may be
defined as:
T
e
-T
w
T
Tw, Tc, Tav
c
=
warm, cold and average temperatures on
an absolute scale
Figure 1.1 shows typical vertical temperature profiles for the
tropical ocean.
Using values of Tw
w
7
=
3000 K(27°C) and T
C
=
2810 K(8°C),
s -
T(-C)
10
20(
301
40
50
60
z(m)
--
Lockheed baseline design assumption
---.---Carribean (Fuglister, 1960)
---Mx
Florida Straits
Hawaii (Bathen, 1975)
Temperature Profiles
Figure 1.1 Examples of Vertical
for the Tropical Ocean.
8
a theoretical maximum efficiency for an OTEC plant of the order of
0.07 may be calculated.
Due to energy losses in the production process,
such as from pumping, friction and imperfect transfer across the heat
exchangers, the actual efficiency of a plant is expected to be of the
order 0.02 to 0.03.
This efficiency is very low compared to conven-
tional steam-electric power cycles which have efficiencies on the order
of 0.30 to 0.40.
In order to produce power in quantities comparable to
conventional plants, an OTEC plant must use very large quantities of
water to exploit its low grade energy resource.
duce 100 MWe, assuming a 2C
For example, to pro-
change in temperature across the heat
exchangers and a plant efficiency of 0.02, a total flow rate of approximately 1200 m3/sec (42000 cfs) is required.
1.2
Current Prototype Designs and the Need for this Investigation
All of the prototype designs currently (1977) being considered
(see Chapter 2) are for free floating plants in
ocean.
the deep tropical
Discharges and intakes, i.e. sources and sinks, may be approxi-
mated as occurring along a single column separated only by some vertical distance.
The very large flow rates involved in plant operation
should be expected to alter the ambient flow and temperature field.
Since the intakes and discharge operate in the same region, it is
possible that the discharged water, which has lost some of the initial
temperature difference, may recirculate directly to the intakes.
Should this happen the temperature difference across the plant would
decrease, which would further reduce the thermodynamic efficiency.
It
is necessary to know if an OTEC plant, based on the prototype designs,
9
will be able to operate without destroying the resource it
draws upon.
1.3
Purpose of this Study
The general purpose of this study is to examine the characteris-
tics of
the flow and temperature field in which an OTEC power plant
would exist to determine constraints on the size and operation of the
plant.
In particular, this study is directed at experimental and
analytical investigations to describe the flow and temperature fields
formed by a schematic OTEC power plant discharging at the interface of
a distinctly stratified ocean with and without ambient currents.
Two
discharge geometries are considered, namely a radial discharge around
the plant circumference and four separate discharges at 90° to each
other.
The results are expected to serve as a first indication as to
what will occur in a more complex design.
The engineering sensi-
tivities and limitations on plant design and ocean baseline parameters,
including maximum and minimum flow rates and discharge velocities,
maximum or minimum temperature differences and depth to the thermocline, can be examined in this schematic framework.
10
CHAPTER II
PROTOTYPE SCHEMATIZATION AND SCALE MODELING PARAMETERS
2.1
Physical Characteristics of the Prototype
Several different designs have been proposed for a prototype OTEC
power plant.
The designs considered in this paper are limited to those
that can be modeled as symmetric vertical columns.
Other designs, not
considered here, rely upon the ambient ocean currents to provide a
continuous stream of warm surface water to the upper intakes.
This
highly asymmetric type of plant must be designed for site-specific
conditions.
The column, or spar-buoy, design does not rely upon the
ambient ocean currents.
This is the design for which plant operating
parameters and ocean baseline conditions are given in Table 2.1.
The
design conditions are taken from studies by Lockheed (1975), TRW (1975)
and Carnegie-Mellon University (1975).
In addition the Lockheed design
has been reduced in power output to 100 MW, equal to the other designs.
The range of these design parameters serves as a base from which a
"standard" condition for the scale model can be drawn.
2.2
Schematization
Designs of this type, and the stratified ocean, can be simu-
lated by a schematization as shown in Figure 2.1.
This simplifies
the scale model and the experimental procedure while retaining the most
important aspects of the external flow and temperature fields.
11
Table 2.1
OCEAN BASELINE AND PLAN DESIGN PARAMETERS FOR VARIOUS PROPOSED PROTOTYPES
TRW
CarnegieMellon
University
X
X
Oceanography
Lockheed
Lockheed
(reduced
output)
Thermocline
60 m
60 m
C200 ft)
(200 ft)
AT Across
18.3°C
18.30°C
22°C
20°C
Plant
(330 F)
(33°F)
(400F)
(38°F)
2.75 m/sec
2.75 m/sec
1.0 m/sec
(5.5 kts)
(5.5 kts)
(2 kts)
Power Output
160 MW
100 MW
100 MW
100 MW
Intake Diameter
X
X
15.4 m
15.4 m
(50 ft)
(50 ft)
Depth
Ambient
Current (max)
X
Plant Design
Plant Diameter at
Level of Discharge Ports
143 m
143 m
60 m
(470 ft)
(470 ft)
(200 ft)
X
1360 m 3 /sec
850 m3/sec
456 m3/sec
368 m3/sec
(48000 cfs)
(30000 cfs)
(16100 cfs)
(13000 cfs)
1800 m3/sec
1133 m3 /sec
456 m 3 /sec
637 m 3 /sec
(64000 cfs)
(40000 cfs)
(16100 cfs)
(22500 cfs)
1.5-2.4 m/sec
1.5-2.4 m/sec
2.4 m/sec
2.13-3.35 m/sec
(5-8 ft/sec)
(5-8 ft/sec)
(8 ft/sec)
(7-11 ft)
Flowrate:
Cold
Warm
Discharge
Velocity
Discharge Depth:
Cold
Warm
X:
88 m
(290 ft)
88 m
52 m
(290 ft)
(170 ft)
46 m
46 m
52 m
(150 ft)
(150 ft)
(170 ft)
Unspecified or Not Known
12
Thermocline
Thermocline
a
4
,i
e)
aU)
4r.I
.rq
44
"o
0
0
44
u
H,-.
44
4i4
"o
a)
4
EH
4
4i
a)
a'4p
C14
EH
"---
----
--------
---
0
-
4,4
.
HCd
p
W
E-4
0,tUU0
C:
0
H -i
Cco
4i)
.
f
'4
Cd
/
-H
w
a)
C -4I
H
a)
H
''
0
U,
-H
Fz4
I
IT
*
JJ
Wa)
bO
H
i IC
.
cd a(d
-4W
-r4
PL4 '--
13
1=
2.2.1
Schematization of the Ocean
Figures 2.1 (b,c) indicate that the ocean is approximated by
two stagnant layers of water of distinctly different densities and
temperatures.
The stratification is stable.
Both layers are assumed
to be isothermal and are separated by an abrupt thermocline.
The adopted schematization is representative of actual ocean
conditions.
Figure 1.1 compares well, in form, with Figure 2.1 (b),
particularly in the upper layer.
profile of the tropical ocean.
Figure 2.2 shows a typical density
This corresponds to the schematic
situation shown in Figure 2.1(c).
layer is pronounced.
The change in density from layer to
Water density in the ocean is a function of sali-
nity as well as temperature.
Figure 2.3 serves to point out that sali-
nity changes at the sites of interest are small and, hence, contribute
little to the change in density over the depth.
The distinction between temperature and density is important
in modeling an OTEC power plant.
A plant generates power by operating
on the thermal difference between layers.
The external fluid dynamics,
however, is controlled by the difference in densities (buoyancy).
Buoyancy in the experiments is regulated solely by temperature.
2.2.2
Schematization of the Plant
Figure 2.1(a) shows that the OTEC plant is schematized to be
a long, narrow, floating cylinder (spar-buoy).
There is one intake port
at the top of the cylinder for the warm water and one at the bottom for
the cold water.
The evaporator and condenser flows are assumed to be
mixed together within the plant prior to the discharge so that the dis-
14
23
25
24
26
27
28
29
aT
10
20
30
40
50
60
aT
=
(p-1)1000
---+--- Carribean (Fuglister, 1960)
---
Florida Straits
----
Hawaii (Bathen, 1975)
Figure 2.2 Examples of Vertical Density Profiles
for the Tropical Ocean.
15
__
(
)
100
200
300
400
500
600
z(m)
-----
0-
--
i
t(Fuglister,
1960)
X
Figure 2.3 Examples of Vertical Salinity Profiles
for the Tropical Ocean.
16
charge volume is the sum of the intake flows and the discharge temperature is the weighted average.
The discharge is assumed to be exactly
at the thermocline so that the jet trajectory will follow the thermocline, neither rising nor falling.
These assumptions of completely mixed flow released exactly
at the thermocline are not strictly accurate.
The actual OTEC designs
being simulated in this study do not mix the warm and cold water within
the plant, nor is the discharged water released in a neutrally buoyant
In some designs (TRW, 1975), however, the separate discharge
position.
streams are arranged close to each other so that mixing immediately
outside the plant can be assumed.
Also, the distance which the discharge
jets would rise or fall is expected to be small compared to the distance
from the intakes.
The qualitative differences of mixed and non-mixed
discharge designs are further addressed in Chapter 9.
Two types of generic discharge configurations are evaluated
in this study:
a)
Radial discharge:
The discharge geometry is assumed as a slot which
completely encircles the plant circumference.
Although none of the pre-
sent designs exhibit this geometry, it is a useful basis for evaluation.
It has an obvious advantage for analytical (cylindrically two-dimensional)
and experimental modeling and it preserves the characteristics of the
more complicated three-dimensional separate discharge design.
This is
possible so long as the radial discharge design retains equality of mass,
momentum and heat fluxes.
b)
Separate discharge:
Four separate jets with rectangular cross-
sections are arranged around the plant circumference at an angle of 90°
17
to each other.
This closely approaches probable (round port) design
conditions with the mixed discharge concept.
Table 2.2 lists the ocean baseline and plant design parameters
with the two generic discharge configurations for a 100 MW "standard"
OTEC plant.
2.2.3
Further Simplification for Experimental Purposes
Even though the schematic prototype greatly simplifies the
description of the external velocity and temperature fields a further
modification has been made in order to facilitate the experiments.
Because of its relative shallowness, the upper layer is of primary
interest because of the potential for recirculation.
As a first
approximation, the discharge jet geometry can be taken as symmetric with
respect to the ambient abrupt thermocline.
This restricts the model
to the upper layer only with a half-jet discharge of different density,
see Figure 2.4(a).
This schematization reduces the total depth of water
required for a physical model and eliminates the need to provide a
carefully stratified ambient environment.
Finally, the model of the upper layer is inverted relative to
the prototype, see Figure 2.4(b).
This measure eliminates the wall
friction that would occur if the discharge jet were at the floor of the
model basin.
On the other hand, the frictional effects on the inverted
"surface" are considered to be negligible due to the small velocities
of the intake flow.
By inverting the model the effective direction of
gravity has been reversed.
In order to maintain the proper sense of
18
A.
B.
Ocean Baseline
Thermocline depth
H
50-100 m (average = 70 m) (210 ft)
Ocean Current
u
0-2.8 m/sec
r
23 m
a
(0-4 kts)
Plant Design
Plant Radius
Intake Flow
0
(75 ft)
3
Qi
Intake Radius
500 m /sec
(17000 cfs)
15 m
(50 ft)
(20°F)
Intake Depth from
Surface
Temperature Difference
between Discharge
and Upper Layer
AT o
11°C
Density Difference
between Discharge
and Upper Layer
APo
0.003
(i)
gm/cm3
(.19 lb/ft
3)
Radial Discharge Configuration
Discharge velocity
u
1.5-3.0 m/sec
(5-10 ft/sec)
Port half height
ho
.62-1.25 m
(2-4 ft)
Port half area per
quarter section
Ao
45.6 m 2
(490 ft 2 )
(ii) Separate Discharge Configuration (Four Parts):
Discharge Velocity
uo
1.5-3.0 m/sec
(5-10 ft/sec)
Port half height
ho
3.8 m
(12.5
Port width
b
5.9-11.8
Port half area
Ao
22-45 m2
Table 2.2:
m
ft)
(19-39 ft)
(72-148 ft 2
)
Prototype Ocean Baseline and Plant Design Parameters for
100 MW OTEC with Two Generic Mixed Discharge Configurations.
19
c I,,
1
I
\
n
N'
\
Orl~~
\o
4
I..,
X
hr
cu
X
CLa
Q-
1-
Q)
.
m
'v
O
VL
QJ
U) o0
o_ o
.
fI
4JU
P )
}4
ur
'
I
t II
I i
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'I
.
,)
Ii
s-I
Eo_
,-,
v
U)
E-4
20
F4
-
U)
3
buoyancy the sign of the temperature difference is also inverted.
The
model discharge water is warmer, rather than colder, than the ambient
upper layer water.
2.3
Model Scaling Parameters
To retain dynamic similitude between the prototype and a scale
model it is necessary to know which effects will dominate the flow and
An accurate scale model will reproduce
temperature fields in both.
those characteristics which are most important in the prototype situation.
In order to determine what the controlling features are in both the model
and prototype, it is necessary to consider the ocean background and engineering parameters.
Table 2.3 lists the relevant physical variables that
determine the external flow and temperature field generated by a schematized plant.
The 20
listed variables involve five basic dimensions:
mass, M; length, L; time, T; heat, J; and temperature,
the Buckingham
.
According to
w -theorem, 15 independent dimensionless groups can be
formed from these variables.
Table 2.4 lists one set of dimensionless
**
groups that may be useful in this study.
*19 for the radial discharge configuration.
**An alternative form of a densimetric Froude number uses the
square-root of the discharge area, Ao (see Table 2.2). This "modified
discharge Froude number"
u
p A 1/2) 1/2
(
IF
P o
where
=
Ao
hob
o
for separate jets
2-r h
for radial jets (choice of quarter
section to compare with four
separate jets)
eliminates the geometry effect of the discharge port and is useful for
the comparison of discharge designs (Stolzenbach, et al., 1972; Jirka,
et al., 1975; but with slight differences in the definition of discharge
area).
21
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CO
C)
C)
4
rl
'4u
re
a)
a
II
I
a
4-
P-d
a
Cd
II
0-
I
co
o0
4-i
co
4i
4-J
a
C)
4a
4-
)
m
R-
a
a)
0
rU
H
bo U
bL)
ca
cC
Cd
10 H) Ico 4N .
p1
4
ra
¢
4-
rl0
11
Ca
aa
m
0l
jr
j
O
22
C
0
~a
0
u
a4
H v
II
II
O
a11
0
"
CJ
0
Ei
au
0
q~
ro
4rJ
a)
C
0
IC)
U
U
U
U
r4
H
"d
O
r4
o d
i
-0
0
0
.- 1
C
En
CL.
C
b0
-
U 44
a)
)
O (x
a)
co
Q
0
a
4>
E-
) -,
trl
Cd
a
a)
0
If
,
.r-I
II
II
II
4
k
Cl
a)
41
"
Q
cE
-4
u
1.
Discharge Densimetric Froude Number
(IFro for radial jet)
(IF
O
0
Ap
(g P
1/2
ho)
for separate jet)
0.
2.
Qi
Intake Densimetric Froude Number (IFi)
Apo
(g
P
51/2
H)
4u h
3.
Discharge Reynolds Number (R
e)
4.
Discharge Weber Number (W)
uo(H)
5.
Heat Loss Parameter
0 0
/2
K
UoPcp
6.
Turbulent Prandtl Number (IPr )
7.
Viscosity Ratio
8.
Density Difference Ratio
Ap/p
9.
Thermal Expansion Ratio
BAT /p
C/K
e/v
O
u H2
10.
a
Ambient Current Flux Ratio
Qi
11.
h /H
Discharge Geometric Ratios
12.
ho b
13.
14.
00o (separate jets)
Intake Geometric Ratios
15.
Table 2.4
hlH
ri/hi
List of Non-Dimensional Parameters for Schematized OTEC
Conditions
23
Given that the same medium is used in the prototype and the
model it is not possible to retain equality for each of the dimensionless parameters.
The model has been scaled to preserve geometric simi-
larity and densimetric Froude numbers.
The model is undistorted and at
a scale of 1:200 to the schematic prototype (Table 2.6).
Densimetric
Froude similarity preserves the buoyant mixing process, which is the
most important characteristic in determining the external flow and
temperature fields (Jirka, et al., 1975).
The approximate scale of 1:200
was determined by the depth of the laboratory basin (35 cm) in relation
to the characteristic upper layer depth of the schematic prototype (70 m).
This choice of modeling criteria satisfies similitude for nondimensional parameters 1, 2 and 11 through 15 listed in Table 2.4.
Parameters 8 and 9, the density difference ratios, are identical in a
system stratified only by temperature.
This ratio is made similar by
proper choice of the discharge temperature.
The model scale must be large enough so that the other dimensionless parameters are nearly satisfied or describe effects which are
negligible in both the prototype and the model.
Table 2.5 lists the
values of the non-dimensional parameters based on the design values
given in Table 2.2 for the schematic prototype and for the experimental
model.
The prototype is operating in the high discharge Reynolds
number range.
The model Reynolds number cannot be made the same since
24
1.
IFr
(radial)
IF
(separate)
o
*
IF
o
Prototype
Model (1:200)
13.5
7-120
7.8
3.8-70
6.0
3.0-43.
2.
IF.
0.012
0.012-0.034
3.
IR e
1.76x107
3100-12400
4.
W
Not Applicable
4.5-290
3xlO- 5
1
5.
Heat Loss
Not Applicable
8.
Apo/p
.003
0.0015-0.003
10.
Ambient Current
Flux Ratio
0-27.4
0-1.4
11.
h/H
0.018
0.009-.018
(radial)
(separate)
12.
h/ro
.054
0.054
(radial)
0.054
0.027-.054
(separate)
0.167
0.167
(separate)
.32
.32-.64
13.
h/b
14.
hi/H
0.86
0.86 (not varied)
15.
ri/hi
0.22
0.22 (not varied)
Table 2.5
Values and Ranges of Dimensionless Parameters for
OTEC Prototype (see Table 2.2) and Model (at a
scale of 1:200)
25
the modeling criteria and receiving medium are chosen.
It has been
shown that for
ŽRe 1500
(2.1)
fully turbulent flow of the discharge jet can be assumed (Jirka, et al.,
1975).
Under these conditions the characteristics of the jet mixing
zone can be taken as Reynolds number independent and thus similar.
Furthermore, if a sufficient turbulence level is maintained in the jet
mixing zone, it can be assumed that the Prandtl number (6) and viscosity ratio (7) are also similar for model and prototype.
The region
outside the jet mixing zone is characterized by considerably less
turbulence and possibly laminar conditions.
However, the dynamic
similarity of heat and momentum transfer in this region is of lesser
importance due to differences in time scale.
The Weber number is not applicable in the prototype since
surface tension plays no role in the discharge flow field.
Weber number (Table 2.5), based on a value of
sufficiently large,
The model
o = 71 dynes/cm, is
= 18, to indicate that surface tension has little
influence on the model flow field.
Another parameter, the heat loss ratio, is also not applicable
to the prototype situation.
-4
K = 6.8 x 10
For the model, the ratio, using a value of
2
cal/cm -sec-°C, is sufficiently small, K/uopci= 3 x 10
5
,to
indicate that most of the heat discharged from the model remains in the
water and little escapes to the atmosphere within the scale of the
experiment.
26
The model values for the dimensionless parameters listed in
Table 2.5 show the variability with respect to the "standard" conditions
given for the choice of the schematic prototype.
In general, the model
was operated over a certain range of the parameters to encompass various
design and operating conditions.
Different discharge and intake densi-
metric Froude numbers were generated by varying the choice of volumetric
flow rate and discharge geometry (as indicated by the variability of the
discharge design parameters, Table 2.6).
The intake design parameters
were maintained the same because the intake geometry can be expected to
have only secondary effects on the type of intake sink flow generated.
The equivalent prototype dimensions for the "standard" 100 MW OTEC
plant (scaled at the ratio 1:200) which formed the base case for the
experimental program is illustrated in Figure 2.5 for the two generic
discharge geometries.
It was not possible to examine the entire range for the ambient
current flux ratio in the model because the experimental current generation system was limited in its pumping capacity.
27
Model:
1:200
Ocean Baseline
H
0.35 m (not varied)
0-.01 m/sec
a
Plant Design
u
0.10-.80 m/sec
0
.00317-.00635 m
h (radial)
O
.0191 m (not varied)
(separate)
b (separate)
.0296-.0592 m
r
0.114 m (not varied)
o
4.4xlO 4-17.7xlO
Qi
m3/sec
ri
0.076 m (not varied)
h.
1
0.030 m (not varied)
AT
5.5-11°C
o
0.0015-.003 g/cm 3
Apo
Table 2.6
Range of Model Ambient and Plant
Design Parameters
28
,-l
Ci
c;
c)
a
C)
U)
· ~
I
0 um0
0
0
'*e
a4
a)
(l
-O
C
CJ
II
'r
c 0 ,e
C))
C
u
,--I
co
cO
'4
U4
0
ci
o
a)
a)
p~co
rX
;'
hM
a
Ca
29
a
Cd
CHAPTER III
EXPERIMENTAL LAYOUT AND OPERATION
3.1
The Basin
The physical experiments were conducted in a 12.2m x 18.3m x
0.35m (40' x 60' x 14") basin located on the first floor of the Ralph
M. Parsons Laboratory for Water Resources and Hydrodynamics at M.I.T.
The basin is equipped with a cross-flow generating capacity of velocities up to 1 cm/sec for the depth at which the experiments were
conducted.
A 4.57 m (15 ft) long plexiglass window was installed in a
wall of the basin for this experimental program.
This allows visual
observation and photography of the velocity field close to the wall.
3.2
The OTEC Model
Figure 3.1 is a cross-sectional view of the cylindrical plexi-
glass model used to simulate the upper portion of an OTEC power plant.
Figure 3.2 is a photograph.
Warm water enters through hoses at the
top of the plexiglass model into bays separated by thin walls.
hose discharges into a separate bay.
Each
plate with small holes drilled in it.
The water passes through a thick
This is to dissipate excess
energy in the flow and break up jets that may form inside the model
which would disrupt the uniform discharge distribution.
Passing
through the holes in the energy dissipator plate, the water impacts
on the lower lip of the upper chamber.
30
The water flows around the lip,
__
INFLOW
PIPE
I
10cm.
(4 n.)
.MIXING
BAY
DI SSIPATOR
PLATE
Variable
T
SPACER
,
1,
30
(12
2 3 cm(9 in.)
Figure 3.1 Schematic Diagram of the Plexiglass
Model.
31
Figure 3.2 Photograph of the Plexiglass Model.
32
through the discharge slot and then into the basin.
The discharge slot
is designed to contract as the water flows away from the center of the
model in such a way as to maintain a constant velocity while the water
is inside the model.
This measure helps to maintain uniformity of the
discharge flow.
The model has the facility to vary the discharge slot configuration by inserting spacers into the model.
Spacer "rings" of dif-
ferent thicknesses allow for discrete changes in the radial discharge
slot height.
The discharge can also be changed to separate ports by
blocking portions of the circumferential
gular ports can be represented.
slot.
In this way rectan-
These two basically different configu-
rations are referred to as "radial: and "separate" jet discharges.
The intake is at the bottom of the plexiglass model.
enters through the bottom into the lower chamber.
Flow
From the lower
chamber the water passes to the upper chamber inside a sleeve surrounded
by the warm water bays.
From the inner
portion of the upper chamber
the water is withdrawn through a suction hose.
The plexiglass model consists of two symmetric halves so that
it can be attached flat to the plexiglass window for the wall set-up
or bolted together and placed in the center of the model basin.
with both configurations were performed.
Runs
Experiments with the half-
model attached to the window make the basin effectively twice as big
as those conducted with the full-model in the center.
model runs the wall is taken to be a plane of symmetry.
For the half-
33
The full
model tests were conducted primarily to insure that wall friction is
not important in the half-model experiments.
3.3
The Discharge and Intake Water Circuits
Figure 3.3 is a schematic diagram of the water flow circuits
and cross-flow generating system.
Constant temperature must be maintained in the discharge flow
to simulate an OTEC plant at steady state.
This is accomplished by
mixing cold tap water with hot water that has passed through a steam
heat exchanger.
A mixing valve adjusts the relative flows of hot and
cold water to achieve the desired density.
From the mixing valve the hot water flows to a 0.21 m3
(55 gal.), 3.0 m (10 ft) high constant head tank.
This helps to provide
a constant pressure in the delivery to the model, but more importantly
damps out potential short period temperature fluctuations inthe hot
water system.
The hot water is pumped from the constant head tank through a
flow meter and control valve to a manifold.
The manifold has eight
valves with connecting hoses to each separate bay in the upper chamber
of the plexiglass model.
The flow rate can be regulated by the valves
on the manifold to make the flow from the plexiglass model uniformly
distributed.
The discharge temperature is monitored at the model.
The intake circuit draws water from the basin through the
center of the plexiglass model.
The water is withdrawn by a pump,
measured by a flow meter and controlled by a valve.
ture is monitored at the model.
34
The intake tempera-
E
I .
I~
J~
V
I_
a
cn
4)
ac
0
E2
E
co
.r-I
u
a)
c·r
x
.r-C)
to
0.j
I0
4
-
0
U
35
3.4
Temperature Data Acquisition, Processing and Presentation
The bulk of the information collected from the physical experi-
ments is temperatures at various places in the external flow field.
100
thermistor temperature probes with a time constant of 1.0 seconds were
used.
Two probes were placed in the hoses feeding the discharge.
or four probes were used to monitor the intake temperature.
Three
The remaining
95 or 94 were mostly fixed to a vertically traversable frame to monitor
the temperatures found in the external flow field.
Figure 3.4 shows the
five probe arrangements used in the experiments.
In the stagnant experiments the probes were set up as to
capture the important parts of an anticipated temperature field.
Radial
experiments had probes arranged along rays extending out from the model.
Each probe had a counterpart at the same radius on other rays.
Separate
jet experiments were set up to measure centerline temperature along with
a few temperature cross-sections (for jet width determination).
Experiments with an ambient current had probe arrangements
meant simply to cover the entire basin. As with the stagnant experiments,
probe densities were higher near the model where the major temperature
variations occurred.
A digital electronic volt meter with the capability to scan
100 thermistor temperature probes records the information on a paper
printout and on punched paper tape.
A typical experiment will record
5000 to 6000 temperature readings.
Computer facilities were used to calibrate, present, and store
this large amount of data.
Computer programs were developed to do each
of these tasks.
36
o
o
0
o
.
.
0
-5m.
I;
0
;
0
I;
+5m.
B
Fxp. *9-15
0
.
.
·
o
.
O
o
- 5 m.
-5m.
·
·
·
·
·
.
·
·
.
0
-5m.
I; C,D
Fxp.
·
O
o
+5m.
II; A,B,C,D
#16-24
0
Exp. *25-42
I:
Half Model Experiments
II:
Full Model Experiments
A:
Radial Jet, Stagnant Exp.
B:
Separate Jet, Stagnant Exp.
C:
Radial Jet, Current
D:
Separate Jet, Current
5m.
+5m.
I; A
TEMPERATURE
·
Exp. #49-61
Vertically
o Fixed
Probe
37
Location Maps
Exp.
PROBE KEY
Traversing Probe
Position Probe
+ Fast Response Probe
Figure 3.4
Exp.
A calibration of all probes was done before each of the five
experimental setups (Figure 3.4).
A constant temperature bath capable of
an accuracy of -+0.020 C was used to provide calibrations at 3 different
temperatures over the expected experimental range.
An additional cali-
bration was done before each experiment by using the nearly isothermal
initial condition
existing
in the stagnant basin water.
A computer
program linearly interpolated between calibrations to produce a corrected
data set.
A versatile plotting program ("PLOT" written in PL1) was
developed for presenting the data.
Each of the thousands of temperatures
recorded during an experiment is characterized by 4 coordinates:
spatial ones (cylindrical or Cartesian system) and time.
3
"PLOT" can
produce 2-dimensional temperature maps with any two of the spatial coordinates as axes and the third one set to a desired value.
a plane in the experimental temperature field.
This defines
Any measurement taken on
that plane over a specified time range is printed on the map at its
appropriate location.
Isotherms were easily drawn from these printouts.
PLOT can also produce graphs of temperature versus any one of the 4
coordinates, including time (the remaining 3 being set to a desired value).
Reduction of the data set by time or spatial averaging is an
option of the program, PLOT.
Measurements taken at the same location
during an experiment can be time averaged and outputed as a single temperature.
nates.
Spatial averaging was possible over one of the horizontal coordiThis was very convenient in radial experiments.
Measurements at
the same radius and depth could be averaged and plotted as one temperature
neglecting angular locations.
38
Some data manipulations were easier using the recorded
temperatures directly.
A computer program (ANALYSIS in PL1) was
developed to orderly print the numbers necessary for plotting a vertical
temperature profile at each of the horizontal probe locations.
Using
the output of this program any vertical profile can easily be drawn by
hand.
Time and spatial averaging
of the data is possible within this
program as well.
The data set, supporting calibrations, and probe locations
for each experiment were stored on magnetic tape.
3.5
Steady State Determination
The value of any data taken rests heavily on how nearly it
approaches the steady state condition that would exist in an experimental
basin without boundaries. Experiments were conducted specifically to determine what time window was available to take good measurements.
A stationary vertical column of ten temperature probes (at
different elevations) was used to measure a vertical temperature profile.
The probes could be scanned in about 5 seconds and thus provided a nearly
"instantaneous" vertical temperature profile.
An experiment consisted of the measurements of four columns
of 10 probes each.
They were scanned at least twice a minute.
In this
way a detailed time history of the vertical temperature profile was
obtained at four locations.
39
3.6
Direct Recirculation Measurements with Fluorescent Tracer
The short-circuiting of warm discharge water into the model
intake is termed recirculation.
It can be quantified by the volume
fraction of discharge water in the intake flow.
Temperature measurements
of the intake water can indicate if any of the hot discharge water is
recirculating.
However practically, this method is of no value if the
temperature rise is 0.050 C or less.
Temperature non-uniformities in the
ambient water can overshadow temperature rises
f this magnitude.
Thus
the minimum recirculation reliably measureable from temperature data is
0.05/ AT
o
= 0.0045 (AT
O
= 11°C) or 1/2 %.
A more accurate measurement was attainable through the release
of a fluorescent dye in he discharge water.
A fluorometer (Turner
Model III) allows concentration measurements down to 1 part per billion
(ppb).
Experiments could be run with discharge concentrations of as
much as 50,000 ppb and basin background concentrations of less than
30 ppb.
A dye concentration of 10 ppb above the background concentration
was distinguishable.
Thus recirculation measurements down to 10 ppb/
50,000 ppb = 0.0002 or 0.02% were possible.
3.7
Fast Response Temperature Measurements
A vertically traversable temperature probe with a time con-
stant of 0.07 seconds was also used in some experiments.
This probe
could provide a continuous plot of the vertical temperature profile by
slowly traversing through the water depth and recording the temperature
variation on a plotter connected to a volt meter.
Temperature fluctuations with time at a fixed point could
40
also be recorded with the same equipment.
In this way a measure of
turbulent temperature fluctuations could be obtained.
3.8
Velocity Measurements
Besides detailed information on the temperature field, it is
desirable to determine the velocity distribution outside of the model.
Velocity measurements are needed before complete understanding of the
fluid mechanics external to an OTEC plant is possible.
In order to be able to observe the vertical velocity structure of theflow field a 4.57 m (15 ft) long plexiglass window was
For the
installed in one of the walls of the model basin (Figure 3.3).
experiments conducted with the half-model, the model was attached directly to the window.
Velocity measurements were made by taking a
A running
sequence of photographs of dye injected into the flow field.
clock in the corner of each picture makes the calculation of the time
of travel straightforward.
Three different methods of introducing the dye into the flow
were developed, each providing different types of information.
injected into the warm water feeder hoses.
Dye was
A visual check of the dye
distribution in plan view would immediately show if the discharge was
properly distributed.
Photographs of the dye front through the window
provided an average time of travel velocity for the discharge jet and
the vertical extent to which the jet had penetrated.
A second method of flow visualization was to coat threads
with dye crystals.
A weight would be attached to one end of the thread.
41
When lowered into the water this would provide a continuous vertical
line source of dye.
By photographing the dye front it was possible to
reconstruct the vertical velocity structure of that point.
The third method of visualization was to drop dye crystals
directly into the water.
Falling through the water the crystals leave
a trail of dissolved dye which moves in the flow field.
closely simulate a vertical line source.
This would
A single, well-defined line
of dye would be produced which would be more exact than the front from
a continuous source.
The major problem with dropping dye crystals was
that they did not always start to dissolve until they had fallen some
distance through the water.
This carried them completely through the
discharge jet, providing no information in that region.
3.9
Experimental Procedure
A standard procedure for performing the experiments was
established.
The same procedure was followed for every run.
The model basin was filled with water and stirred the day
before the experiment was to be conducted.
the water would be isothermal and stagnant.
This was to ensure that
Before the experiment
would be started, the warm water was turned on and run through a bypass to a drain.
This allowed the discharge temperature to stabilize
before using it in the experiment.
During this time the temperature
probes were scanned to establish the ambient temperature and obtain
a probe calibration.
When the warm water had reached the desired
temperature the model intake was turned on.
42
If the basin water level
was correct, the discharge was started and dye was injected into the
warm water feeder line to determine if the discharge distribution was
uniform.
After the dye front in the basin passed the last probe
temperature scans were started to define the existing temperature
field.
The data taking procedure for temperature is tailored to the
flow situation.
Close to the surface the discharge jet introduces
significant turbulent temperature fluctuations.
Therefore, close to
the surface several scans at each level are taken so the temperature
readings can be averaged.
After one level has been scanned the frame
supporting the probes is moved to a new depth.
Thirty seconds are
given for the probes to come to the new equilibrium and then the scans
are repeated.
Approximately twenty minutes were required to scan the
entire basin depth.
During the experiment, photographs of dye strings, crystals
and injections into the model are taken to provide information on the
velocity structure.
Readings with fast probes are also taken.
Three days or less were generally required to calibrate,
plot, and store the results of an experiment.
Procedures and results
were continuously evaluated as the experimental program progressed.
43
CHAPTER IV
EXPERIMENTAL RESULTS:
RADIAL DISCHARGE
AND STAGNANT OCEAN
4.1
Run Conditions
A series of laboratory experiments was conducted to inves-
tigate the external flow and temperature fields induced in a stagnant
ocean by a schematic radial OTEC plant characterized by the parameters
varied over the ranges presented in Chapter II.
The dimensional parameters and a description of each experiment appear in Table 4.1.
Table 4.2 contains the dimensionless gover-
ning parameters (as discussed in Section 2.3).
Starting with the
"standard" base case (100 MW) the experiments were designed to evaluate
sensitivities by increasing the flow rate and discharge velocity and
decreasing
AT
and slot height, h .
There were four series of experi-
ments with a radial discharge into a stagnant basin.
The first series
(Experiments 1-6 and 8) covered the desired range of parameters.
The
other three series were to support and detail the results of the first
series.
The second series consisted of Experiments 26, 30, 31, and
33.
They were
done with the full model in the basin center. The main
purpose was to determine if the wall in the half model experiments
had a significant effect on the flow field.
44
Table 4.1
DIMENSIONAL PARAMETERS AND EXPERIMENT DESCRIPTIONS
STAGNANT RADIAL JET EXPERIMENTS
Run
Type Model
Half
1
2
Full
Qi
[cm3 /sec]
X
884.
442.
3
X
X
4
X
5
6
X
X
7
X
8
X
26
X
X
X
X
X
30
31
33
44
45
46
X
47
48
X
49
X
50
X
51
54
55
56
57
58
59
X
X
X
X
X
X
X
X
60
X
X
61
X
884.
1770.
884.
1770.
1770.
1770.
884.
884.
884.
884.
884.
884.
1770.
884.
1770.
1770.
1770.
1770.
1770.
1770.
1770.
1770.
1770.
884.
884.
884.
u
o
[cm/sec]
19.4
9.65
19.4
38.8
38.8
77.5
77.5
77.5
19.4
19.4
19.4
38.8
19.4
19.4
38.8
38.8
77.5
77.5
77.5
77.5
77.5
77.5
77.5
77.5
77.5
19.4
19.4
19.4
A
h
o
TD
TA
Ap x 103
of
[cm 2 ]
[cm]
[0 C]
[°C]
[gm/cm3 ]
11.4
.64
.64
.64
.64
.32
.32
.32
.32
.64
.64
.64
.32
.64
.64
.64
.32
.32
.32
.32
.32
.32
.32
.32
.32
.32
.64
.64
.64
34.6
34.6
27.9
33.4
33.5
33.1
33.1
38.1
23.9
26.6
28.0
28.2
29.2
30.6
30.8
30.8
30.9
32.6
32.4
32.4
26.4
22.9
35.0
33.9
34.8
35.0
34.7
34.6
23.6
23.0
22.2
22.4
22.7
22.2
22.7
22.9
13.9
13.5
15.3
15.4
17.2
18.8
19.1
18.9
19.2
21.6
21.3
21.3
23.3
23.1
24.6
24.7
25.0
24.9
24.8
24.8
3.23
3.37
11.4
11.4
11.4
5.7
5.7
5.7
5.7
11.4
11.4
11.4
5.7
11.4
11.4
11.4
5.7
5.7
5.7
5.7
5.7
5.7
5.7
5.7
5.7
5.7
11.4
11.4
11.4
Types
Measurements
A
1.45
A
A
A
A
A
A
A
A
A
A
A
3.11
3.05
3.04
2.92
1.37
1.96
2.68
2.83
2.86
2.82
2.97
2.96
2.99
2.97
3.01
3.03
3.05
0.77
0.051
3.25
2.70
2.95
3.06
2.97
2.90
__
B
B
B
B
B
A,E
A
A,C,D
A,E
A,E
A,C,D
A,D
A,D
A
__
A,C,D
A,D,E
Description Key
A.
B.
*
Overall Temperature Field
C. Velocity (Dye Photograph)
Steady-State Determination
D. Fast Temperature Probe
E. Fluorescent Dye Recirculation
Discharge Temperature:
** Ambient Temperature:
Average of % 25 Individual Measurements Taken
Periodically during the Experiment.
Spatial Average of Basin Temperature Measurements
just before the Experiment
45
Table 4.2
DIMENSIONLESS GOVERNING PARAMETERS
STAGNANT RADIAL JET EXPERIMENTS
Run
Type Model
Half
Full
F
IF
IF.
h /H
r /h
o
1
X
5.91
13.62
.069
.018
18.
2
X
2.88
6.64
.034
.018
18.
3
X
8.81
20.31
.102
.018
18.
4
X
12.06
27.81
.140
.018
18.
5
X
14.49
39.73
.071
.0091
36.
6
X
29.02
79.58
.141
.0091
36.
7
X
29.62
81.22
.144
.0091
36.
8
X
43.13
.211
.0091
36.
118.3
26
X
7.59
17.50
.088
.018
18.
30
X
6.49
14.97
.075
.018
18.
31
X
6.32
14.57
.073
.018
18.
33
X
14.94
40.97
.073
.0091
36.
44
X
6.35
14.64
.073
.018
18.
45
X
6.19
14.27
.071
.018
18.
46
X
12.40
28.59
.143
.018
18.
47
X
14.67
40.23
.071
.0091
36.
48
X
29.40
80.62
.143
.0091
36.
49
X
29.15
79.93
.142
.0091
36.
50
X
29.07
79.72
.142
.0091
36.
51
X
28.99
79.50
.141
.0091
36.
54
X
57.42
157.5
.281
.0091
36.
55
X
615.1
1.091
.0091
36.
56
X
28.08
77.00
.137
.0091
36.
57
X
30.77
84.38
.150
.0091
36.
58
X
29.44
80.73
.143
.0091
36.
59
X
6.09
14.04
.070
.018
18.
60
X
6.19
14.27
.071
.018
18.
61
X
6.24
14.39
.072
.018
18.
224.3
46
The third series (Experiments 44-48) was steady-state
determination tests with profile measurements taken at radiuses of 2
and 3.5 meters.
Only experiment 44 was with a full model.
four experiments were for a half model at the wall.
were conducted at the wall.
The other
Experiments 49-61
The temperature field measurements were
reduced to provide time for other types of measurements.
Fast response
temperature probe measurements, dye photography, and dye recirculation
sampling were all done in this series.
4.2
Discussion of Results
Observations made thorughcut the experimental series allowed
a qualitative picture of the flow field to be formed. Three flow zones
were easily recognizable (Fig. 4.1). The jet and intake flow zones are a
consequence of the model's discharge and intake ports.
The return
flow zone supplies the volumes of ambient water necessary for jet
entrainment and the intake flow.
For experiments with deep, high
entrainment jets, return flow velocities approached the magnitude of
the jet zone velocities.
4.2.1
Temperature Field Measurements
Figure 4.2 is a typical example of data provided by a
radial stagnant experiment.
The temperatures represent the spatial
average of all probes of equal radial distance from the center of the
model.
The temperatures at different depths are not taken simulta-
neously because the probe frame takes some time to traverse to a new
level (at a speed of 30.5 cm/min).
47
However all temperatures were
4A
-
-
1
I
_
_ -
-
4
o
HciI:
z
,,.I
to
4-i
0n
a,)
E!
*i-4
I
3
p
U)
0.
x
w
4J
P>
a)
3
Cd
I'll
--4
co
a)
co
.,,
Cd
Cd
4-'
co
0
'0W
co
4i
t
0
,--
\
W
vI
-,
zL
0N
\\
-H
-4
I\--
rXo
.41
HO
bo
l/
/
I4:
_
I
i6
48
_
PQ
,- 1-
oZ
o
E
P.O
Q)
E4
E-4
-
v-b4J
.1rd
Ha
PN
.,
44~
0
6
c
c
o
Iqcon
49
n
o
E
recorded within the time span in the experiment which approximates
steady state.
Multiple scans were taken at the levels closer to the
surface to derive time-smoothed values that averaged out the local
turbulent fluctuations.
Due to time and spatial averaging, the
values presented close to the surface represent a compilation of 20
data scans while the lower levels represent 5 data scans.
Figure 4.3 is a typical plan view of the isotherms for the
level scanned closest to the surface (i.e. 0.5 cm below the surface).
This figure shows that spatial averaging is, indeed, necessary because
the discharge flow is not perfectly radial.
This deviation from ideal
radial flow can be caused by a non-uniform discharge distribution and
by wall friction, in the case of the half-model experiments.
Full
model experiments 31 and 33 indicate that the radial nature of the
flow field is reasonably well represented.
Radial averaging of the
data should minimize the error from not obtaining perfectly radial flow.
Appendix A.1 presents the radially averaged plots of the
normalized isotherms, given in per cent, for each experiment.
4.2.2
Steady-State Determination Results
Typical results of these experiments are illustrated in
Figure 4.4.
Temperature scans were taken with high frequency to ob-
serve the jet front pass through the measurement location.
Temperature
readings then settled down to a steady state with only turbulent
fluctuations.
The end of the steady state was usually signaled by
the uppermost probe that had been at ambient temperature.
50
It would
4
0
C
U
z
C)
co
X
0
-
UZ
C'4
C
lJ
Dc~
r.k
51
X
1ci
"i
11
0~~
0
IS
4.J
C4)
.--
cr0C
:
o
M
C)1
-E
0
I- U
I
a
w
aw
52
_
_CI
I
_
_
___ll__·I__I__W_____·-L·LLa-··-·--·
D
w
U
1
4
begin to show some heatup effect.
The mean temperature of the probes
in the jet would subsequently show a steady rise in temperature.
It
seems likely that the heated water building up at the edges of the
basin forces an intermediate warm layer to flow back toward the model.
This is supported by visual dye observations.
Table 4.3 is an attempt to determine the beginning of
steady state and the onset of rising temperatures.
The end of the
steady state regime was not a perfectly abrupt phenomena.
ment had to be used in the construction of the table.
Some judge-
In general, a
mean temperature deviation of more than 0.1°C (at either the surface
or in the middle of the jet) was considered the cutoff point for the
steady state.
Exp.
IF *
Configuration
Model
Discharge
Flow Rate
(cm3 /sec)
Approximate Period of
Steady State @ 3.5 m
Start
End
43
15.30
Sep. Jet
Full
884.
9 min.
23 min.
44
6.35
Radial
Full
884.
11 min.
25 min.
45
6.18
Radial
Half
442.
10 min.
34 min.
46
12.39
Radial
Half
884.
8 min.
33 min.
47
14.66
Radial
Half
442.
9 min.
32 min.
48
29.42
Radial
Half
884.
7 min.
33 min.
Table 4.3
Experimental Steady State Durations
53
4.2.3
Velocity Measurements
Velocity measurements using the techniques described in
Chapter III were taken at fixed points.
Figure 4.5 is a photograph of
a typical dye string used to construct a velocity profile.
Velocity
measurements obtained by these methods provide only local information.
Since the flow field may have some non-uniformities, as it is linked
to the non-uniform temperature field, it is difficult to generalize
this local data to an average velocity distribution for the entire
induced flow field.
A typical velocity profile which shows the jet
forward velocity in the surface layer and a lower, uniformly distributed return flow in the lower layer is shown in Figure 4.6.
Other
data on velocity measurements (taken in Experiments 51, 56, and 60)
is summarized in Appendix B.
4.2.4
Fast Probe Measurements
Fast probe measurements were taken during 4 experiments
(Experiments 51, 58, 60, and 61).
The purpose was to examine the
turbulent temperature fluctuations at two locations for conditions
similar to experiments 1 and 6.
Sets of measurements were taken at
two different distances from the model (0.9 m and 2.1 m).
A vertical
profile and stationary measurements were taken at each location as
described in Chapter 2.
The results and some preliminary analysis appear in
Appendix C.
A notable feature is the dominant slower temperature fluc-
tuations near the bottom of the jet as compared to the higher frequency
turbulent fluctuations in the upper portion of the jet.
54
Figure 4.5 Photograph of Dye String Velocity
Measurement.
55
O
C
L
O
4-J
C
~Ln
pr)~
I
-H
k
a4
q4
In
u)
,-
O
.
--
4 :a
+
a)
S-~
~ ~
x
-/
N
a
a)
X
C4-i>4
C
Ca
a
cad
Ed
rl
·
a)
d
>.
O
a.
0
m
0
C(
-)
cn
4-i
0
-
~J
3
O
N
0 W
GQ
56
__
i__ _1_1______111_____I__^-
-^-1___
CN -
I
E
u
F')
4.2.5
Recirculation Measurements
All the low Froude number and low discharge experiments
(1 to 5, without dye
measurement;
61 with dye measurement) showed
no indications of direct recirculation either in the temperature
data, the visual dye observations, or the dye measurements within the
time span for which the steady state condition persisted.
In all
these experiments, little vertical penetration of the surface layer
was observed (less than 40% of the total depth) as shown in
Appendix A.1.
On the other hand, experiments with high Froude numbers
and high discharges (6 and 8, without dye measurement;
44, 54, and 55
with dye measurement) indicated a tendency for recirculation based on
the following observations:
1)
Dyed discharge waters appeared to penetrate much
deeper over the water column (more than 50% in
Experiments 6 and 49;
Experiments
2)
over the entire depth in
8, 54, and 55).
The intake temperature increased by a small
amount ( <.020 C for Experiment 8) within the
first 30 minutes of simulation.
3)
The dye concentration measurements showed small
increases in concentration (However, no concentration
rise was observed in Experiment 49, the repeat of
Experiment 6).
The recirculation measurement results
are summarized in Table 4.4.
57
%
Recirculation
*
Exp.
61
I
o
50 min.
Type Measurement
..
-
6.2
30 min.
20 min.
10 min.
IF
--
Values
(4.
0. %(t.02%)*
0z(1.02%)
0% (02%)
Fluorescent Dye
( 2%)
0%( 2%)
0%(t2%)
Intake
Temperature
%)
0% (4.%)
0% (+4.%)
Fluorescent Dye
.3% (+. 3%)
.3%(+.3%)
Intake
Temperature
Fluorescent Dye
6
29.0
0.
49
29.2
0.%
8
43.1
.3%(+.3%)
54
57.4
.5%**(+.08D .7%**(+.1
.16%(±.044
55
224.
12%(+2.%)
18%(+ 3.%)
*
**
20%(+3.%)
22%(+3.%) Fluorescent Dye
The estimated possible error of
measurement is in parentheses
Recirculation was increased by an
intake flow rate 10% over the
correct value
*** This discharge was negatively buoyant
AT = -.20 C
Table 4.4
Recirculation Measurements
In principle, recirculation could be caused by the intake flow directly entraining warm discharge water or by the model basin
being too small to allow steady state to be achieved.
It was shown in
Table 4.3 that steady state jet behavior can at least be expected during
a 15 minute
period
of the experiments.
The fact that experiments
8, 54,
and 55 measured recirculation during that period implies that the first
kind of recirculation mechanism was observed.
58
---
_____ _____1___^___^__1_1__11__·__1__1___1__1
__..
The mechanism of recirculation was observable in Exp. 8 by
introducing dye into the discharge water in short bursts.
The patches
of dyed discharge water could be observed through the basin window as
they mixed.
At a radius of approximately 0.75 m, the dyed water came
in close proximity to the bottom.
ccasionally, large eddies could be
seen to mix locally all the way to the bottom, break off the jet, and
become part of the.return flow zone.
Faintly dyed water was observed
to enter the model intake occasionally.
The phenomena was very unsteady and appeared to occur randomly as far as time and angular location were concerned.
The approximate
radius of occurrence (about two water depths) though never changed.
At
larger distances from the model, a fairly well defined two layer flow
was apparent from the temperature data.
The layers were approximately
of equal thickness.
4.3
Experimental Correlation of Discharge Behavior
Figure 4.7 is a graph of stable jet dilution as a function
of the modified Froude number IF
0
((
0
U
g
r
(see footnote in Section 2.3)
ho]
(4.1)
1/2) 1/ 2
Iglfh
This governing parameter is chosen in order to more easily relate to
the separate four-port discharge geometries.
S,
The stable jet dilution,
is defined to be the total volumetric flow rate at a radius, where
the temperature is changing slowly ("stable region"), divided by the
59
25
S
s
20
15
10
5
0
20
10
30
4Q
IF
Figure 4.7
Stable Jet Dilution vs. Quarter Module "Port"
Discharge Froude Number, Stagnant Experiments
60
discharge flow rate.
Using the similarity assumptions presented in
Chapter VII, it is possible to linearly relate the local jet dilution
to the ratio of discharge temperature difference
and the local jet
surface temperature difference
AT
s
s
where I
AT
I
0
c
(4.2)
IG
and IG are integration constants defined in Chapter VII.
Figure 4.7 shows that the majority of the experiments
exhibit a linear relationship between S
S
and F
O
.
This is consistent
with the theory of single port buoyant surface jets (Stolzenbach and
Harleman*, 1971; Jirka, et. al.*, 1975) for which a correlation has
been developed:
S
S
= 1.66 F
(4.3)
O
The two highest Froude number points, corresponding to runs
6 and 8, do not agree with Equation (4.3), giving further indication
that the mixing process is limited and recirculation may occur in
these tests.
Figure 4.8 is a graph of the maximum jet half-temperature
depth (i.e., the depth where the temperature difference above ambient
is one-half the centerline value) normalized by the square-root of
*
t
I
The result
in these reports is given as Ss = 1.4 F
2¼ F * due to differences in the port area definition.
where Fo
0
61
=
A
.
1U-
8-
h
5Tmax
6-
4-
21
30
-I
/
2
0
I
I
I
10
20
30
I
40
IF
Figure 4.8
Normalized Maximum Half-Temperature Depth vs.
Quarter Module "Port" Discharge Froude Number,
Stagnant Experiments
62
the discharge area, (2 ro ho) , as a function of F
.
The data point
for run 8 is included even though the jet's temperature field appears
to extend all the way to the basin floor.
A linear trend in the data
is appraent and the data correlation with the theory for single port
buoyant jets is as good as that for S..
Assuming a polynomial vertical
temperature profile (Abramovich, 1963), the relationship between maximum half-temperature depth and F
h
0
is (Jirka, et. al., 1975):
*
.ST max = 0.22
FA
F
(4.4)
o
It is significant that the data point for run 6 shows an
upward deviation from the general trend set by the other experiments.
This seems to indicate that the jet extends deeper due to the effects
of depth limited mixing.
In summary, Figures 4.7 and 4.8 show that the greater the
discharge Froude number, the more turbulent mixing (i.e., higher
dilution) and the deeper the jet until some limiting value of the
Froude number is realized where the effects of the confined layer limit
entrainment and recirculation occurs.
4.4
Extreme Case Experiments
An important exterior flow phenomena of interest to OTEC
operation is recirculation.
But the experimental schematization of an
OTEC plant in operation showed some tendency for,but no significantly
high values of,this phenomenon over the chosen parameter ranges.
63
Therefore, experiments 8, 54, and 55 were conceived with lower discharge
temperature differences and higher discharge Froude numbers.
Higher
values of recirculation were obtained, but because of the small temperature difference, the experiments lost some of their applicability to
the mixed discharge configuration of an OTEC power plant (see Chapter 9).
Each of the three experiments showed some degree of measurable
recirculation (Table 4.4).
with F
0
= 230, AT
0
=
-
Only the most extreme experiment (Exp. 55
.2 C) had a really sizable recirculation(which
even appeared to show a transient ris).
The dyed discharge waters were
just reaching the basin boundaries when experiment No. 55 ended after
fifty (50) minutes.
Because of the decrease of buoyant spreading, this
time of travel is approximately two to four times as long as in other
experiments.
It must be kept in mind that the recirculation values of
these experiments have no implications for OTEC operating with a mixed
discharge configuration.
However, the experiments show a definite
possibility for recirculation in a non-mixed discharge scheme.
In this
case, smaller discharge temperature differences and higher discharge
Froude numbers would realistically occur in the evaporator discharge
jets.
This aspect is further discussed in Chapter 9.
64
CHAPTER V
SEPARATE JET
EXPERIMENTAL RESULTS:
AND STAGNANT
5.1
DISCHARGE
OCEAN
Run Conditions
The separate jet discharge configuration of a schematic
OTEC plant in a stagnant ocean was examined in a series of laboratory
experiments.
The induced external temperature and flow fields were
investigated as discharge parameters were varied over the ranges
presented in Chapter 2.
The dimensional parameters and a description of each
experiment appear in Table 5.1.
Table 5.2 contains the dimension-
less governing parameters (as discussed in Section 2.3).
Starting
with the "standard" base case(100 MW), the experiments were designed
to evaluate sensitivities by increasing the flow rate and discharge
velocity and also decreasing ATo and the port width, bo.
Half model
experiments with the OTEC model attached to a wall consisted of
jets directed at 450 from the wall and 900 from each other.
two
The full
model in the basin center discharged four separate jets directed at
900 from each other.
The experiments can be divided into three sets.
initial set of six half model experiments (Exp.
desired range of parameters.
65
The
10-15) covered the
The remaining two sets were for
Table 5.1
DIMENSIONAL PARAMETERS AND EXPERIMENT DESCRIPTIONS
STAGNANT SEPARATE JET EXPERIMENTS
u
Qi
A
h
[cm2]
[cm]
11.3
11.3
11.3
1.9
1.9
1.9
o
o
b
o
T
TA
Ap x 103
Type Model
[ cm3 /sec
] [cm/sec]
Full
Run Half
884.
1770.
884.
884.
1770.
1770.
19.6
X
884.
19.6
41
X
X
X
884.
884.
884.
43*
X
884.
39.2
19.6
39.2
39.2
10
11
12
13
14
X
15
X
X
X
X
X
35
37
39
39.2
19.6
39.2
5.64
5.64
78.4
78.4
5.64
11.3
5.64
11.3
5.64
5.64
1.9
1.9
1.9
1.9
1.9
1.9
1.9
1.9
[cm]
[°C] [C]
[gm/cm3 ]
3.26
5.9
35.0
5.9
34.8
23.9
24.1
5.9
3.0
3.0
27.4
21.8
33.3
22.2
33.5
22.3
3.0
27.7
22.2
1.38
5.9
3.0
5.9
30.6
18.9
2.97
30.5
18.8
2.96
28.9
2.92
3.0
3.0
28.4
29.5
16.4
16.3
17.8
3.18
1.40
3.12
3.15
2.77
2.82
Table 5.2
DIMENSIONLESS
GOVERNING PARAMETERS
STAGNANT SEPARATE JET EXPERIMENTS
Run
10
11
12
13
14
15
35
37
39
41
43
Type Model
Half Full
IF
o
IF
IF.
.068
.138
.104
.070
.139
.210
.071
.072
.072
.074
.073
0
X
5.97
7.93
X
12.10
9.10
14.50
28.98
43.56
16.06
12.08
16.19
32.35
48.64
8.31
16.64
8.38
X
X
X
X
X
6.26
X
X
14.90
6.31
X
X
15.40
15.3
17.19
17.08
66
I
h /b
.32
.32
.32
.64
.64
.64
.32
.64
.32
.64
.64
h /H
r 0 /b 0
.054
1.93
1.93
1.93
3.86
3.86
.054
.054
.054
.054
.054
.054
.054
.054
.054
.054
3.86
1.93
3.86
1.93
3.86
3.86
verification purposes only.
One set (Exp. 36, 38, 40 and 42) consisted
of full model repeats of two half model experiments (Exp. 10 and 13).
The last set was a single steady state determination test (as described
in Chapter 3) for a full model (Exp. No. 43).
The breakdown of steady
state behavior appears to be caused by a build up of mixed water at the
basin confines.
The duration of steady state would then appear to
depend on the overall mixing characteristics and be independent of the
particular generic design.
Experiment 43 showed that the separate jet
discharges reached a quasi-steady state of similar duration found in
radial experiments.
5.2
Discussion cf Results
The flow and temperature fields of these experiments are
by nature 3-dimensional and more complex than the radial discharge
case.
Figure 5.1 illustrates the observed flow field in a typical
experiment.
The jet and intake flow zones have similar counterparts
in the radial discharge case.
However, the return flow has been
divided into two separate zones:
1.
lateral return flow; 2.
bottom
return flow.
The two return flow zones can be easily distinguished.
bottom return flow zone is made up of entirely ambient water.
The
It
mainly supplies the intake flow zone and entrainment flows to the
underside of the jet.
the
The lateral return flow zone, existing between
ets, has an associated temperature increase caused by eddies
67
*o.
FL
PL
In
\
%.X
IH
bo
Fr
<
X~~~~F
<
m
\
l~~~~3,
4,
o
-I
L
fl
`,
\ X \
Cc
m
C
'U
K0L
N
CY-N~~~~~
I
c
\U
,1,
I
0
N
r
0
rm
a
0
--
-
r t
t
I
E
L
i
jI
LL
I
a
/
0
0
-r4
u
4i
I
co
EDL
k
III 0,.0
I
I
I
U
Il)
I
I
I
-p
=
t01
·r
Ur!
0
C)
~~~~~~~~~~~a)
r4
I.-
a
,
I
I
-r4
--T-T=.
4
*
L
4,
nbO
L
4
'o
c
>
co
P4
IL
)w
f
0)
0
'4
a,
:
o
((
w
LL w)
r.
as
C
I'
ax-
o NC-)
o
No
0
11
m
C
_E1-
-1
3-I
Ei
ff
ll
;t
Ct
'
68
-
r
breaking off the jet and mixing into the return flow.
This zone has
a depth approaching the deepest penetration of the discharge jets.
Thus, the entrainment flux at the lateral jet boundaries is made up
of a mixture of ambient water and already mixed water which recirculates laterally.
Particularly for cases with jet penetration approach-
ing the total layer depth, the lateral return flow dominates the bottom
return flow.
In these cases, it makes up the major portion of the
jet entrainment flow and, possibly, of the intake flow.
This appears
to be the probable mechanism for near field recirculation of this
generic discharge design.
5.2.1
Temperature Field Measurements
Figures 5.2 - 5.4 are typical examples (Exp. No. 11) of
Figures 5.2
the data provided by a separate jet stagnant experiment.
and 5.3 are centerline sections of the two jets of a half-model
experiment.
Isotherm plots of this type appear in Appendix A.2 for
the experiments in this series.
The temperature (C)
at depths above
10 centimeters represents the time averages of up to four actual
measurements (all taken within two minutes of each other).
lower depth temperatures are single scans.
The
This data collection
procedure averaged out turbulent temperature fluctuations in the
jet to obtain a better estimate of the mean temperature.
As in the
radial experiments, temperature measurements at different depths
occurred at different times.
But all the temperatures were taken
within the time span in the experiment which approximates steady
69
0
4.J
t4
CC
0
t
t
U)
0*
i
4
0
4-©
o
)
,I
4-)
C)
C-
C
I-I
-
r-
.i-
02
70
C)
T01O
c(N
/
a0 -
(
Q
N
N
N
N
N
cD
N
nl
N
.r-
'-
1, .
C 1
N
t
-
U.
O
CM
-L
L
r
Ei
Nr
sI.4*i\ C
o
o
N
N
.
~,
O
o
N
,
N
.I'
,
N
N
N
O
0i
',O '
N
N
-
°
|
CM
A
(J
d.
No
N
N'
L
q
d
.~
~ D~a-'
,
c
,
oo
0
/ CU
N
~
N
X
z
N
1
_
0
N
N
~~~
o'~
'
']-F
u
8
tO
"'"
B
N
a
w
a
i
;
71
OQ
C
E
Ui
c
hlN
"
N
N
1
-J
state.
Figure 5.4 is a typical plan view of the temperature field
for the scan taken closest to the surface (in this case 1.0 cm below
the surface).
The two jets produce very similar and symmetric temper-
ature fields.
It is important to note that the temperature field
gives no indication of the actual jet width.
Even probes nearly half-
way in between the jets, register a temperature rise above the ambient
(24.1°C).
Observation of moving dye patches showed that these temper-
ature probes were substantially outside of the jet velocity field.
This jet structure is due to the lateral return flow mechanism as
discussed earlier.
5.2.2
Steady-State Determination
The "steady state" time interval of these experiments can
be approximated by the previous results for radial jets.
The best
approximation would be from a radial jet that dilutes to the same
degree or that has the comparable discharge densimetric Froude number,
*
IF.
o
One steady state test though was done to examine the time
varying behavior of the jet lateral return flow zone.
Columns of
probes (as described in Chapter 3) were placed at 2.0 and 3.5 meters
from the model (full model experiment).
There were two columns at
each radius, one was located on a jet centerline and the other halfway between two jets.
Figure 5.5 shows the time varying behavior
72
cr
rl
4
3
Q.
X
Eto
C)
1-4
C.
Cr3
c w
r
A
c-)
:
LA
E 0 +
j
L0 u
73
C
of temperature for a depth of 2 cm at each location.
The jet lateral return flow has a shorter steady state
period than the jet flow.
The period of 10-25 minutes is a reasonable
approximation for this test.
periods.
Half model experiments should have longer
It is important to note that the temperature rise in the
lateral return flow zone occurs near the model first and progresses
out from there.
The chronology of Figure 5.5 clearly shows that the
temperature rise in this zone is not caused by discharge jet waters
reaching the basin boundaries, being turned, and flowing back toward
the model (basin boundaries are
5.2.3
7 m. from the model).
Recirculation Indications
There were no direct fluorescent dye studies done for the
separate jet design.
Intake temperature records showed no definite
indications of recirculation in any experiment.
The two experiments
with the highest Froude numbers and discharges (Exp. 14 and 15),
however, indicated some tendency for recirculation based on the
following observations.
1)
Dyed discharge waters appeared to penetrate well over
50% of the water column.
At least one of the jets in
This lateral "shedding" of eddies is a characteristic feature of
buoyant surface jets and has been documented in earlier investigations by Adams, Stolzenbach and Harleman (1975).
74
Ij
TV
E)
<
m
C
Ii
.
Q
i
c
rE
J
· 4-J
4
o
4
(U
,4
ru
s
+'
LL
I
C
-
-'
C
(U
I
<
-
tI
m
c,
~
4)
<
a)
m
-
a
m
t-
12..
Cl
I
I--.1 0
(a,
I
I
LI
L,
o
LP
Ln
01
t)
0ttnl
.4
I--W xa-
m ql--D
w
A.
75
.ri
Exp. 15 appeared to reach the basin floor.
2)
Temperature profiles taken in the interaction return
flow zone indicated a significantly deeper penetration of this zone.
In Exp. 14, the penetration was
over 50% of the water column.
The zone in Exp. 15
appears to extend to 90% of the water column.
Experiment 15 may very well have experienced a small amount
However, the intake temperature actually fell
of recirculation.
0.030 C due to temperature non-uniformities in the basin prior to
the experiment.
5.3
Experimental Correlation of Discharge Behavior
5.3.1
Jet Behavior
Figure 5.6 is a graph of stable jet dilution as a function
of the modified Froude number, F0
.
The stable jet dilution, S , is
defined to be the total volumetric flow rate at a distance where
temperature is changing slowly ("stable region"), divided by the
discharge flow rate.
Using the similarity assumptions presented in
Chapter VIII, it is possible to relate the local jet dilution to the
ratio of discharge temperature difference and the local jet surface
temperature difference:
s=
s
1.5 AT /AT
o
c
Because surface heat loss can become important in the far
76
field region, it is best to measure the stable dilution (when
temperature measurements are involved) at the beginning of the farfield zone.
An estimate of the distance to the transition from near
field to far field is given by Jirka et al. (1975).
t
=12.6 F oio
0o
(5.1)
0
The stable dilution was estimated from surface temperatures measured
at approximately this distance from the discharge point.
Measure-
ments in each of the jets (2 for the half model and 4 for the full
model) were averaged to obtain the experimental value.
Figure 5.6 shows that the majority of experiments exhibit
a linear relationship between S
S
and
0
.
Furthermore, it is
consistent with the correlation developed by Stolzenbach and Harleman
(1971) for single port buoyant surface jets.*
S
5
= 1.66 IF0
(5.2)
The Stolzenbach-Harleman relation appears to slightly
overestimate the dilutions (based on temperature measurements) for
the lower Froude number experiments.
This trend is explainable
because the relationship was developed for a jet entraining ambient
water, while the jets in the present experiments entrained some heated
water from the lateral return flow zone.
This would tend to decrease
The numerical value of Eq. (5.2) differs due to changes in the Froude
number definition.
77
0
15
40
S s = 1.66 F0*
30
Ss
20
10
I
IL
0
10
30
20
I
40
Fo
Figure 5.6
Stable Jet Dilution vs. Quarter Module
"Port" Discharge Froude Number, Stagnant
Fxperiments
78
the effective jet dilution.
The two highest Froude number experiments (Exp. 14 and 15)
Two explana-
have dilutions considerably below the predicted values.
tions are possible.
As was noted in Section 5.2.3, the jets in these
Possible inter-
two experiments penetrate deep into the water column.
actions with the bottom probably limited dilution.
Also, the calcu-
lated values of distance xt to the far field were beyond the confines
of the experimental basin (8.76 m. and 13.16 m.).
This means that the
jets possibly had not yet reached their stable dilutions at 8.0 meters
where the temperature-dilution measurements were taken.
Figure 5.7 plots maximum experimental values of jet halftemperature depth (i.e., the depth where the temperature difference
above ambient is one-half the centerline value) normalized by the
square root of the discharge area, (h b ) .
00
as a function F
o
.
These depths are plotted
The linear trend in the data is apparent and
extends to the large Froude number experiments as well.
One cannot
hope to extend the linear relationship much beyond Exp. 15 because
(Section 5.2.3) the maximum jet depth in that experiment was clearly
the experimental basin floor.
Assuming a polynomial temperature
profile (Abramovich, 1963), the relationship between half temperature
depth and
F
et. al., 1975)
is (Jirka,
h
.5T max
o0
oh
0
79
= 0.18
F
(5.3)
d f
1II
/
'15
8
14,u
L
Ilr
6
h.Tmax
4
2
39
100o
n
/
~
0
~
10
~~
1*
.
20
30
FFiure
5.7
Normaized
M
imum Half-Teperature
Figure 5.7
40
Deth
Normalized Maxirum Half-Temperature Depth
vs. Quarter-lodule "Port" Discharge Froude
Number; Staanant Experiments
80
As seen in Figure 5.7, the data fits this relationship well.
5.3.2
Characteristics of the Lateral Return Flow
This section presents information derived from temperature
profiles taken within the lateral return flow zone at distances of
1.0 m. and 4.0 m. from the OTEC model.
The zone properties are
averaged within an experiment (measurements were taken between different
jets and between jets and the wall in half model experiments) and
plotted as a function of F
o
tion parameter, b/ro,
.
The experimental range of the separa-
does not appear to affect the data trends.
Figures 5.8a and 5.9a are plots of surface temperature in
the lateral return flow zone.
higher jet dilutions.
Higher Froude number experiments achieve
Hence, the surface temperature in the return
zone is less because the jets contributing the heated water are more
diluted.
The centerline jet temperatures at the same distance from
the model are plotted for comparison.
It is of note that (especially
for low Froude number experiments), the return flow surface temperature is higher at 1 m. than the jet centerline temperature at 4 m.
Apparently, some jet eddies are breaking off and entering the return
flow between 1 m. and 4 m. from the model.
Figures 5.8b and 5.9b present an average half temperature
depth in the zone for each experiment.
It is notable in Figure 5.8b
that the lateral return flow zone is much deeper than the jetzone for
large Froude number experiments.
81
Such conditions seem to indicate
Q2
aTo
aT.0
0.1
nn
10.
0.
20.
30.
4Q
F
Figure 5.8a Comparison of Jet Centerline and Lateral Return
Flow Surface Temperature Data at
X
=
l. Om.
-
25.
x
20.
7--1
0
10.
i
+
5.
0
*
*
v.
1
a
0a
-
I
I
20.
10.
FO,
l
30.
40.
0
Figure 5.8b
Comparison of Jet Centerline
and Lateral Return Flow Half Temperature
Depth Data at X = 1.Om.
82
I; -,A=0.32
t
+
;
/bo 0.64
L.R.F;ho= 0.32
LRF h/b= 0.64
0.2
AT
6~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I
hTo
__ _
0.1
_ _
_
0 ._
_
_
0~~~~~~~~~~~~~~~~~~~~~
_
_
__
_
_
_
_
I
aO~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~+
-0
I
I
x
0.0
O.
I
I
20.
10.
40.
30.
Figure 5.9a Comparison of Jet Centerline and Lateral Return
Flow Surface Temperature Data at X = 4.0m.
z
l
-
-
25.
20.
b
------------
x
l
.--
-
I
--
.
TO
_
15.
o
h.5T
x
10.
(cm.)
5.
o
x
+
n
I
I
C
A.
10.
A
·*·
II
20.
30.
0F
Figure 5.9b
Comparison of Jet Centerline
and Lateral Return Flow Half Temperature
Depth Data at X = 4.0m.
83
40.
·
i/bo
~
0.32
=
o
t;
+
L.R.F; h/b = 0.32
x
LR. Fj
h/b o =0.64
h/b
= 0.64
that part of the entrainment flow through the bottom of the jet comes
from the lateral return flow zone.
Conceivably, for increasing densi-
metric Froude numbers, the intake flow would be supplied by this flow
zone too.
However, this possibility was not observed within the range
of the experimental program.
84
CHAPTER VI
EXPERIMENTAL RESULTS:
6.1
OCEAN CURRENT CONDITIONS
Run Conditions
Experiments with the model discharging into an ambient
velocity field were conducted both with radial and separate jet
The model ambient current flux ratio range presented in
geometry.
Table 2.5 is not as broad as the range to be found at a typical
prototype site.
The model current generation system was limited
to average velocities of less than 1.0 cm/sec. for the water depth
used in the experiments.
At the approximate length scale of 1:200,
this model velocity simulates only 0.25 knots in the prototype.
Another difficulty with the ambient current experiments
was the non-uniformity of the vertical and lateral velocity distribution.
Figure 6.1 is a typical ambient velocity profile in the vertical
direction.
Constant velocity from surface to bottom was not possible
because of bottom friction and local non-uniformities in the flow.
Lateral uniformity of the flow was also not guaranteed, partly
because of the slow flows involved.
Uniform withdrawal across the
depth at the downstream end of the basin was difficult to achieve.
In the thermally stratified system induced by the model preferential
withdrawal from one particular layer could occur.
85
0
5
10
z(cm)
15
20
25
30
0.25
Figure 6.1
0.5
0.75
1.0
1.25
Typical Ambient Vertical Velocity Profile,
Cross-Flow Experiment
86
Ambient
Velocity
(cm/sec)
Table 6.1 lists the dimensional operating and design
The dimensionless governing parameters
parameters for the experiments.
appear in Table 6.2.
Both radial and separate jet discharge configura-
tions were tested with half and full model experiments.
The half model
tests for separate jets were limited to a single orientation due to
symmetry considerations.
The two jets had to be discharged 450 and
1350 to the current direction.
With the full model, any orientation
of the separate jets was possible.
In addition to the orientation
used in the half model, tests were done with the orientation having
jets directed upstream, downstream and 900 to the current.
6.2
Experimental Results
A simple, unified description of the flow fields observed
is not possible.
Certain observations though are helpful in under-
standing the nature of the experiments.
Except for the deeply mixing case of Exp. 17, the model
operation formed essentially a two layer flow system over most of
The ambient current dipped down below the heated surface
the basin.
The upper layer was formed by the
layer to form the lower layer.
discharge and retained some of the mixing features of the stagnant
experiments.
tion.
The relative thickness of the layers depended on loca-
In general, the upper layer was thicker for strongly mixing,
higher Froude number discharges.
87
Table 6.la
DIMENSIONAL PARAMETERS AND EXPERIMENT DESCRIPTIONS
RADIAL DISCHARGE EXPERIMENTS WITH A CURRENT
Run
A
Run Type
Model
[3
Half
[cm /sec]
Full
[cm/sec]
Type
2
o
[ ]
2]
[cm
AP x 10
TA
A
3]
u
[gm/cm I
[]
a
[cm/sec]
__
21
X
1770.
77.5
5.7
31.9
20.5
3.04
22
X
884.
38.8
5.7
29.5
18.7
2.66
1.11
23
X
884.
38.8
5.7
29.1
18.2
2.65
1.27
24
X
884.
19.4
11.4
31.7
19.8
3.13
.97
.75
32
X
884.
19.4
11.4
27.9
15.7
2.74
.94
34
X
884.
38.8
5.7
27.8
15.5
2.75
.94
Table 6.lb
SEPARATE JET EXPERIMENTS WITH A CURRENT
Run
Type Model
Half
Full
Orientation
0
45
u
Qi
0
3
clm /sec
cm/sec
T
A,
2
cm
o
TA
Ap x 103
3
°C
°C
gm/cm
u
a
cm/sec
17
X
X
1770.
78.4
5.6
25.4
20.3
1.19
.41
18
X
X
1770.
78.4
5.6
30.5
20.2
2.68
.94
19
X
X
884.
39.2
5.6
30.7
20.3
2.71
.86
20
X
X
884.
19.6
11.3
33.5
23.0
3.01
.88
36
X
X-
884.
19.6
11.3
30.3
19.4
2.78
.95
38
X
X
884.
39.2
5.6
30.0
18.6
2.82
.93
40
X
X
884.
19.6
11.3
28.6
16.5
2.81
.93
42
X
X
884.
39.2
5.6
28.7
16.5
2.82
.91
Jets directed parallel and at 90° to current direction.
Jets directed at 45
and 1350 to current direction.
88
Table 6.2a
DIMENSIONLESS GOVERNING PARAMETERS
RADIAL DISCHARGE EXPERIMENTS WITH A CURRENT
Run
Type Model
Half Full
IF
0
h /H
IF.
r
o
/b
o
u H2/0
a
21
X
29.0
.141
.0091
36.
.52
22
X
15.5
.076
.0091
36.
1.54
23
X
15.5
.076
.0091
36.
1.76
24
X
6.0
.070
.018
18.
1.34
32
X
6.4
.074
.018
18.
1.30
34
X
15.2
.074
.0091
36.
1.30
Table 6.2b
SEPARATE JET EXPERIMENTS WITH A CURRENT
Run
Type Model
Half Full
Orientati.on
00
°
45
IF o
IF
A=h /b
o
h /H
r /b
oo
a
H 2/Qi
17
X
X
46.9
.226
.64
.054
3.86
.28
18
X
X
31.3
.151
.64
.054
3.86
.65
19
X
X
15.6
.075
.64
.054
3.86
1.19
20
X
X
6.2
.071
.32
.054
1.93
1.22
36
X
X
6.5
.074
.32
.054
1.93
1.32
38
X
X
15.3
.073
.64
.054
3.86
1.29
40
X
X
6.4
.074
.32
.054
1.93
1.29
42
X
X
15.3
.073
.64
.054
.054
3.86
1.26
89
A persistent feature of the radial half model experiments
with a current was that the portion of the flow discharged directly
upstream separated from the wall, as can be seen in Fig. 6.2.
Thus,
the presence of the basin wall in the half model radial experiments
could not be totally neglected.
This separation was, of course, absent
for the full model tests.
The temperature field near the surface for separate jets was
The jets were
essentially similar to the stagnant case (Figure 6.2).
bent by the current until they were directed downstream.
A lateral
return flow zone of some form existed between each pair of jets (or a
jet and the wall for half model experiments).
The return zone between
jets pointed 450 to the upstream direction was mainly ambient water.
But even in this zone, the
model
surface
temperature would rise near the
due to eddies breaking off the discharge jets.
None of the experimental currents were sufficiently strong
to confine the model discharge widthwise within the basin.
water reached the far basin wall in each experiment.
Heated
In some discharge
configurations, heated water nearly reached the upstream basin wall as
well.
6.2.1
Temperature Field Measurements
The data presentation is essentially the same for the
current experiments as for the stagnant experiments.
No spatial
averaging is possible, however, because there is no intrinsic symmetry
90
F
j
oo
rz
PI
0
I,4
0
co
U;
0
s-4 CW
1I
!i
.
°
91
^
01
4I
.
cN
O
O
N
Nu
-
N
,i
0O
(N
Nu
CM
N
N
N~
Nu
CM
NM
Nu
0k
N~
U)
IX
O
N
NU
NV
N
IN
N
N
CM
C)
Cr)
M
N
(V)
C
CM
N
OH
CM
(1)
N
C\
Cr)
N
0
H
11
N
Cr)
N-
N
col
rP4
S
Q4 *,i
j
*rl.,
Pi
H
eq p
*c -d
E
Ci
0
In
6(N
60o
*
N
CM
6
0
cm
N
In
6i
0
92
-H
in the flow and temperature fields of the half model.
Multiple scans
The
were taken near the surface to smooth out turbulent fluctuations.
upper level temperatures represent the average of 4 data scans while
the lower layer temperatures are taken from only one scan.
Appendix A.3 presents the normalized isotherms, in per cent,
for longitudinal transect AA and lateral transect BB shown in Figure
6.2 in addition to the surface plans.
Full model experiments actually
have two longitudinal transects AA on either side of the model.
Only
one is presented in Appendix A.3.
Figure 6.3 is a plan view of the uppermost level with the
current moving from left to right.
radial.
The discharge configuration is
Isotherms have a distorted shape due to the current effect.
The model basin does not contain the discharge jet since some of the
surface isotherms extend to the wall opposite the model.
As previous-
ly mentioned, the discharge jet splits away from the upstream wall for
this half model experiment.
6.3
Experimental Correlations of Discharge Behavior
Figure 6.4 is a plot of an average jet dilution as a function
of IF at a downstream distance of 5.5 m.
o
Readings of several surface
temperature probes at this distance (5 or 6 depending on whether it is
a full or half model) were averaged.
A dilution was calculated assuming
a uniform surface temperature (the average value just mentioned) across
93
4
Ss
: 1.66 *
3
V18
S5
17
21
1
0
10
20
I
I
30
40
_,
50
F*
* half model.,radial jet
o full model].,
radial jet
* half model.,separate jet
+full model,separatejet
Figure 6.4
Stable Downstream Dilution vs. uarter Module
"Port" Discharge Froude Number, Ambient
Current Experiments
94
the basin width.
The assumption of vertical, polynomial temperature
and velocity similarity profiles (as suggested by Abramovich, 1963)
allowed a dilution calculation.
The line in Fig. 6.4 is the previously used stable dilution
relation, Eq. (5.2), which in principle holds for the stagnant case
only.
However, the data for the experiments with currents appear to
generally agree with the relationship.
The lack of agreement for the
higher Froude number is similar to the stagnant cases and may be caused
by recirculation effects.
This agreement between the current and stag-
nant cases seems to point to the primary conclusion that within the
experimental range the OTEC effluent mixing is dominated by the
discharge design and not by the current speed.
Some secondary effects
which may be caused by the presence of the current are discussed in the
following.
The slightly lower dilutions could also be caused by several
other factors.
For example, the interference on the discharge flow
field by the experimental basin boundaries might reduce dilution.
This effect on full model experiments should be greater than on their
half model counterparts.
Also, the "blocking" effect, in terms of the
upstream density wedge, may cause lower effective dilutions.
Discharge
waters directed initially upstream will be turned and carried downstream
by the current.
These discharge waters then served as sources of
entrainment for downstream directed jets, lowering their effective
95
dilution.
Figure 6.5 is a graph of the half temperature depth at the
point of maximum, near field
Shown for comparison is the
penetration of the model discharge.
relationship
(Eq.
5.3),for maximum penetration of a single jet in stagnant ambient
waters.
The generally good agreement between the current case data
and the stagnant jet correlation shows that the ambient current did
not greatly affect the near field mixing process.
The increased
scatter of data points over those obtained in stagnant experiments is
in part due to the difficulties involved in creating and withdrawing
a uniform current at either end of the model basin (Section 6.1).
96
1Z.
v
17
10.0
8.0
h.Tmax
6.0
4.0
2.0
V-.u
0
10.
20.
30.
40.
50.
'Fo
* half mode,radial jet
° full model,radial jet
Figure
6.5
v half
model, separate jet
+full
model, separate jet
Maximum Half Temperature Depth vs. Quarter
Module "Port" Discharge Froude Number;
Ambient Current Experiments
97
CHAPTER VII
RADIAL JET THEORY WITH STAGNANT CONDITIONS
7.1
Method of Analysis
In addition to the physical experiments, theoretical models
(based on the conservation
equations for mass, momentum and heat)were
constructed to examine the external velocity and temperature fields
produced by OTEC plants with radial discharge geometry (this chapter)
and separate discharge geometry (Chapter 8).
In both cases, a stagnant
ambient condition is assumed, which appears to represent a critical
case regarding the discharge mixing and the potential for recirculation.
A fully analytical treatment of the complete flow field is difficult
because there are different flow regimes induced throughout the entire
region.
Therefore, the theoretical models divide the external field
into different zones.
Each zone is characterized by different aspects
of the flow field such that some effects dominate and others are
assumed to be negligible.
Simplifying assumptions can reduce the
complexity of the governing equations within each zone in order to
make them more tractable to solution.
Figure 7.1 is a schematic diagram of the zones into which
the analytical model has been divided.
The induced discharge flow
can be divided into a jet zone and a far field zone.
The induced
intake flow far away from the intake port is accounted for as a return
98
a
H
w
.
In
a)
-J
0
t
o.
c
It
id
c
E
o
cc
0
-4
ur rl
a)
W
L
99
flow.
These zones are not generally well-defined, but, rather, merge
continuously one into another.
A fully analytical treatment of the
entire induced flow field would require that all boundary conditions
linking the different zones be made compatible.
The laboratory experiments indicate that for the design and
operating parameter range describing typical OTEC conditions, the
controlling mechanism for recirculation of discharged water to the
intake is turbulent mixing in the jet zone coupled with return flow
to the plant.
Thus, for a single plant in a stagnant environment,
the stratified flow in the far field apparently has no constraining
effect (in the form of a potential thickening of the mixing layer) on
the near field.
The far field serves primarily as the sink for the
discharged mass and heat and probably need not be considered in the
analysis of near field recirculation.
The plant intake induces part of the return flow; the other
part is caused by jet entrainment.
In other words, a double sink flow
exists close to the plant due to the intake and turbulent jet entrainment.
A detailed accounting for the flow induced by the plant intake
would, thus, require an analysis of coupled turbulent source and sink
flows in a stratified environment.
Analysis and experimental studies
of non-turbulent single sink flows in a stratified regime have been
carried out by Craya (1949) and Rouse (1956) in a stagnant ambient
100
field and more recently by Slotta and Charbeneau (1975) in an ambient
current.
These studies have not considered the presence of entraining
turbulent density interfaces with regard to withdrawal, so, do not apply to the OTEC situation.
The thrust of the following analysis is toward a detailed
treatment of the turbulent jet discharge which is coupled with a
return flow in a confined layer.
A thorough examination of the results
and properties of these jet equations can reveal information regarding
the near-field stability of the OTEC discharge and the potential for
recirculation.
The analysis does not include the far field zone due
to its likely unimportance.
By coupling the jet zone to the return
flow, the most important effect that the intake has on the induced
velocity and temperature fields has been implicitly included.
No
explicit consideration of the turbulent stratified sink phenomenon is
The equations for radial jets with return flow are developed
made.
for both the surface jet, as represented in the experimental program,
and for the interface jet, as typical for the prototype.
7.2
Model for Radial Buoyant Jet Mixing with Return Flow
The jet zone is that region which is dominated by turbulent
mixing.
In this model, the jet zone has been further subdivided into
a zone of flow establishment and a zone of established flow.
101
7.2.1
Zone of Established Flow
The distinguishing characteristic of fully established jet flow
is that the vertical distributions of velocity and temperature can be
That is, the velocity, u, and temperature,
assumed to be self-similar.
AT, distributions for uniformly distributed radial flow can be characterized by equations of the form
ur = u (r)f(z/h)
(7.1)
AT = AT c (r)g(z/h)
(7.2)
The centerline velocity, u r, and temperature difference, AT ,
from ambient are functions of the radial distance, r, only.
The
functions f and g must be fit from experimental data (Albertson, et.
al., 1950; Abramovich, 1963) where h is the "depth" of the jet.
Appendix D presents a derivation of the radial jet equations
with and without return flow for the model situation (surface jet).
A solution for the prototype discharge at a distinct density interface is also presented in Appendix D.
Utilizing the similarity
assumption for velocity and temperature and ignoring dynamic pressure
effects, the vertically integrated jet quantities in Appendix D can
be re-written:
h
u dz
=
ur hI1 - Q
c
o
102
(7.3)
2
U|dz
u
2
(7.4)
M
A.I=
h
Ih
u Tdz - u
C~ J
AT r~~~~
hlG ~
-=
where ur is the velocity relative to the lower layer.
are functions of radial distance only.
(7.5)
u ,
ATc and h
The integration constants are
defined as:
(7.6)
I1 =fJf(n)dn
of 2 (n)dn
12 =-
(7.7)
O
f(n)g(n)dn
IG =
(7.8)
The vertically integrated differential equations for the
conservation of mass and momentum, including the effect of return
flow, for both the surface and thermocline discharge, D.14 and D. 15,
become
Continuity
(1
h
1 dr
103
+
=0.
(7.9)
Momentum
1 drM
r dr
2 drQ
rH dr
+
1
d2
2
_
dr
rH
dh
H2 dr
H
w
H
h
1 dP
p dr
10)
The pressure term is:
dP = I
dz
(7.11)
o
and is derived for the surface and interface jet situations, respectively, in Appendix
D.
The surface and interface discharge jet equations differ in
their treatment of the heat transport.
(i)
Surface Jet
Substituting for the integral quantities, the surface
discharge heat transport equations, (D.16), is:
1 drJ
1
r dr
d
[r
Hr dr
(7.12)
J
I21M
where the integration constant
1
G1
=
g(n)dn
(7.13)
The vertically integrated pressure derivative (ignoring the
dynamic pressure term) in the momentum equation derived in Appendix E.1
is:
104
2
d d [ 3 IG
dr dr
I3IG
d
d 2
3
(7.14)
1
where
G2
=
(n')dn'dn
(7.15)
o r
ii)
Interface Jet
For the prototype discharge jet at the density interface,
the temperature along the centerline is constant.
Under these condi-
tions, the heat transport equation, (D.21), becomes:
AT
dru IG
r
dr
druIG
Hr
dr
aT 2(wlh)T
(Wh)
(7.16)
This is exactly the same as the mass conservation equation
provided IG = 2I1
.
It can be shown that for any anti-symmetric temper-
ature distribution, g(z/h), and any symmetric velocity distribution,
f(z/h) (see Figure D.2) this condition holds:
r1
IG =
0r
f(n)g(n)dn
1
f(n) (2-g(-n))dn
=
f(n)g(n)dn
+
o
=
2
f(n)d-
2 f(n)dn = 21
0
105
(7.17)
Therefore, since the temperature distribution is known, the
heat transport equation is redundant.
The vertically integrated pressure derivative in the momentum
equation for the interface jet is developed in Appendix D.2.
This term
is the same as for a surface jet:
dP
dr
2
d [Hq 33 IG22
M2
dr
M
I3
-]
(7.18)
I1IG
In the surface and submerged discharge cases, the entrainment
velocity, -W1h, is modeled in the same manner.
The rate of entrainment
of water into the jet is assumed to be directly proportional to the
jet centerline relative velocity.
-wlh
= au
(7.19)
Ellison and Turner (1959) have shown that for a buoyant jet
the entrainment coefficient, a, is a function of the local jet
Richardson number, which is the inverse of the square of the local
2
densimetric Froude number, IF
=
L
2
rc
/g
P h.
p
A best fit curve for
their experimental data is expressed by the following relationship
(Stolzenbach and Harleman, 1971):
-Wlh = a exp(-5/FL-w
L ) u rc
106
(7.20)
7.2.2
Zone of Flow Establishment
Equations (7.9), (7.10), (7.12), and (7.20) are the set that
describe the surface discharge jet with return flow found in the model
situation.
Equations (7.9), (7.10), (7.18) and (7.20) form the set
which describe the submerged jet discharged at a distinct density interface as found in the schematic prototype.
In both cases, boundary conditions are necessary in order to
solve the systems of equations.
The region between the discharge
opening and the beginning of self-similarity (fully established flow)
is the zone of flow establishment.
The end of this zone provides the
boundary conditions necessary to determine the solution.
The zone of flow establishment is dominated by the discharge
momentum and buoyancy can be neglected for the discharge conditions
under consideration.
It has been found empirically (Albertson, et. al.,
1950) that for uniform discharge velocity, the length of the flow
establishment region, Lfe, is given by
Lfe = 10.4h
(7.21)
The jet depth, hfe, at the beginning of established flow is determined
from momentum conservation:
h r
hfe = I)(7.22)+
0
Lfe
107
(7.22)
Discharge Parameters
u
Discharge Velocity
Slot Height
0
h
0
Temperature Difference
Plant Radius
AT
O
r
Thermocline Depth
0
H
Integral Coefficients (profile dependent)
I1,
I2
IG, G 1 , G 2
Entrainment Coefficient (profile dependent)
Table 7.1
INPUT INFORMATION TO ANALYTICAL MODEL FOR RADIAL JET
108
Polynomial Profile
(Abramovich, 1963)
Gaussian Profile
(1_ 3/2)2
f (1)
e
-n
2
2
3/2
g(n)
I1
0.45
12
0.316
IG
0.368
0.6
0.2143
0.5
0.0495
0.0682
*
*
=
I1
2
db
dr
db
dr
is derived from experiments,
d
from Abramovich, 1963,
dr
= 0.22 (fit for polynomial profile)
from Albertson, et. al., 1950,
db
dr
= 0.154 (fit for Gaussian
profile)
Table 7.2
FUNCTIONS AND COEFFICIENTS FOR ABRAMOVICH AND GAUSSIAN PROFILES
109
where h
is the plant slot height and r
is the plant radius.
The
centerline temperature at the boundary between the two jet zones is
equal to the initial value, AT
7.3
Solution Method
The set of vertically integrated ordinary differential
equations describing steady uniformly distributed radial jet flow at
both the surface and density interface have been solved using a fourthorder Runge-Kutta routine on a digital computer.
The length and depth of the zone of flow establishment are
computed to provide the boundary conditions for the fully-established
flow region.
The centerline velocity and temperatures are predicted
as is the depth
to half of the temperature
centerline and ambient water.
are calculated.
difference
between
the
The local jet Froude number and dilution
For the cases including backflow, the return velocity
and depth to zero velocity are provided.
The information necessary to solve the equations is listed
in Table 7.1.
The discharge parameters are determined by the situation
being simulated.
The integral coefficients and entrainment coefficient
are dependent upon the choice of velocity and temperature profiles.
Table 7.2 gives the appropriate values for these coefficients for
polynomial (Abramovich, 1963) and Gaussian profiles.
The analysis of the fully established flow region extends
from the zone of flow establishment to the radius at which the local
110
jet densimetric Froude number declines to unity.
At this point, the
jet zone is arbitrarily terminated and the far field is assumed to
begin.
In fact, the transition from jet flow to buoyancy driven flow
is continuous with the importance of turbulent entrainment decreasing
while interfacial friction is increasing.
The model sometimes predicts
unrealistic oscillatory behavior for the jet thickness as the Froude
number approaches unity, indicating the existence of critical stratified
flow conditions.
However, the actual occurrence of critical conditions
would be determined by the behavior of the stratified far field flow
(with interfacial friction) which is not treated in the present
analysis.
7.4
Model Results
The analysis of the model and the examination of its sensi-
coefficient,
r 0O0
, r/h
and H/h0, and the model
0
, has been conducted for the surface jet employing the
tivity to the discharge parameters; Fr
polynomial distribution for vertical velocity and temperature profiles.
This profile was chosen for the cosmetic advantage that a specific
depth for zero relative velocity and temperature difference is predicted.
The profiles and coefficients are those listed in Table 7.2.
The surface and density interface jets exhibit the same
general behavior.
Initially, the jet expands linearly since buoyancy
does not have much effect on damping turbulent entrainment.
As
buoyancy becomes more important (i.e., the local jet Froude number,
111
FL
-
1.0), the thickness increases more slowly until the jet merges
into the far field
(]FL
= 1.0).
late jet properties until F L
entrainment ceases.
The computer
code continues
to calcu-
= 1.0, signifying a stable region where
The surface jet centerline temperature initially
declines very rapidly with distance and then levels off when the stable
jet region is reached.
The density interface jet temperature at the
centerline is constant since it entrains water of different temperatures
equally from both sides of the jet.
7.4.1
Surface Jet (Laboratory Model) Predictions
7.4.1.1
Discharge Parameter Sensitivity
A set of baseline discharge parameters in the practical range
was determined for the purpose of model comparison:
F
= 20, r /h =
o o
r
0
20, H/h
o
= 50.
Figures 7.2a and b show the sensitivity of normalized
centerline temperature, AT /AT , and normalized half-temperature depth,
h 5T/ho, as a function of normalized distance, r /h , to the discharge
Froude number.
Figure 7.2a illustrates that for increasing IF
the
o
For higher
r
centerline temperature decreases at any radial distance.
, the centerline temperature sensitivity decreases close to the
o
plant because jet entrainment is not damped by buoyancy so that the
IF
r
maximum amount of mixing takes place.
increasing IF
Figure 7.2b shows that for
the depth of the jet increases at any radial distance.
o
This reaction is limited by the fact that if the edge of the jet
112
extends to the limit of the confining layer, the similarity assumption
is no longer valid and the model is not applicable.
The formulation
implies that recirculation occurs.
Figures 7.3a and 7.3b show the sensitivity of the jet to
changes in the relative size of the plant, ro/ho.
As ro/ho gets larger,
the flow field undergoes relatively less radial expansion.
ly, in the case r /h
+
, the discharge
For larger r/ho
plane jet.
,
flow becomes
Theoretical-
a two dimensional
there is less of a decline in the center-
line temperature and less sensitivity to a change in r /h .
The depth
of the jet is greater for increasing r /ho, since the radial expansion
of the diverging flow has proportionately less effect.
Figures 7.4a and 7.4b illustrate the effect that different
layer depths have on the same discharge configuration.
For small H/h ,
the jet thickens until it extends over the entire layer depth.
H/h
For
large enough to allow buoyancy to damp turbulent entrainment and
limit the growth of the jet thickness, the predicted centerline temperature and jet thickness are not very sensitive to changes in H/h .
7.4.1.2
Model Coefficient Sensitivity
The model employs essentially one empirically derived
coefficient once the velocity and temperature profiles are chosen.
entrainment coefficient,
, has been derived from a best fit to experi-
mental data (Abramovich, 1963, and Ellison and Turner, 1959).
7.5a
The
Figures
and 7.5b indicate the effect that a small change in a exerts on
113
1.0
0.8
0.6
AT
AT
O1
0.4
0.2
0
100
Fiqurp
7.2e
200
300
400
500
600
700
800
Temperature vs. Distance, Varying
900oo /h
Fr
O
50
45
40
35
h .5T
h
.0
30
25
20
15
10
5
0
100
Figure 7.2h
200
300
400 500
600
700
800
900
r/h 0
Half-Temperature Depth vs. Distance, Varying
Fr
0
114
1.0
0.8
AT
0.6
AT
0.4
'0.2
100
200
300
400
500
600
700
800
900
r/h0
Figure 7.3a Temperature vs. Distance, Varying
r/h
m~0
50
45
40
35
h.5T
ho
30
25
20
15
10
5
100
200 300
400
500
600
700
800
900
r/ho
Figure 7.3b Half-Temperature Depth vs. Distance, Varying rO/ho
115
Fr = 20.
n8
b
r = 20.
ho
0.6
AT
ATL
0.4
I.
I
0.2
30.
H
-40.
h
tiA 1
0.
.
,
20C.
!
400.
.!
.
.
i
600.
·
!
i
800.
1,000.
Temperature
vs./h
Figure 7.4a
Temperature vs.
Distance, Varying H/ho .
0
Fr = 20.
ro
30.
r = 20.
h.
h5T
ho
20Q
-4n
H
-=40.
ho
10.
0.
0.
200.
Figure 7.4b
Varying H/ho
400
600.
800.
1,000.
Half Temperature Depth vs. Distance,
116
1.0
0.8
0.6
AT
AT
0.4
0.2
200
Fjgqir
7, 5a
400
600
800
r/h
0
Temperature vs. Distance, Varying a
50
45
40
35
h 5T
h
0
30
25
20
15
10
5
0
Fioure
200
7. 5b
400
600
800
r/h 0
Half-Temperature Depth vs. Distance, Varying a
117
the centerline temperature and jet thickness.
The model is not overly
sensitive to changes in a.
7.4.2
Interface Jet (Prototype)
The centerline temperature of the radial jet discharged at a
distinct density interface does not change since it entrains equally
from both sides.
The jet thickness responds similarly as a function
of radial distance for the same
F
r
as the surface jet except that
o
buoyant damping of turbulence is much more pronounced.
For this reason,
the interface jets with the same F
lesser thickness.
as a function
as the surface jets expand to a
o
Fig. 7.6 gives predictions of the jet half depth
of IF
, Fig.
7.7 as a function
of r /h
and Fig.
7.8 as
o
a function of H/h .
In general, the same trends are observed as in the
corresponding surface jet plots, however, with lesser penetration depth.
The information of Fig. 7.2b and 7.6 is further summarized
by plotting in Fig. 7.9 the maximum predicted (asymptotic) jet depth,
h
max
, (where h = 1.6 x h T for the "Abramovich" distribution) as a
'.5T
function of F
r
, again for the specific case r /h
o
O
and H/h
o
o
= 50.
The interface jet prediction for maximum jet thickness is
less than that for a surface jet of equivalent F
.
For this combina-
0
tion of parameters, the model predicts the surface jet will expand to
half of the available layer thickness at IF
face
= 25, whereas the inter-
jet will not achieve the same condition until
F
r
50.
This
o
graph indicates that predictions for the prototype flow field based on
118
15.0 C
F= 40.
10.0 _
hoT
ho
5.0 _
50.
no
I
0.0
0
I
50.
100.
I
150.
20.
i
200.
r/h
Figure 7.6
Half Temperature Depth vs. Distance, Varying
IFro; Radial Interface Jet.
119
15.0
100
h.5T
ho
5.0
0.0
0.
50.
150.
100.
/hFialf
T ture
De
Ficjure 7.7 half Temperature Depth vs
Varying
r/h;
o
Radial Interface Jet
120
Distance,
200.
15.0
-
10.0 _
H
h.5T
ho
30.
5.0
0.0
I.
0
I
I
I
50
100
150
200
Wh o
Half Temperature Depth vs. Distance, Varying
Figure 7.8
H/ho ; Radial Interface Jet.
121
hm,
H
v.v
0.
20.
40.
60.
80.
Fr
Fioure 7.9 Normalized Maxim-mJet Thickness vs.
Discharqe Densimetric Froude Number
for Surface Jetand Interface Jet
122
__
__I_________
100.
a surface jet model will be conservative with regard to recirculation
of discharged water to the plant intake.
At the same time, it must be
kept in mind that the model predictions (for both surface and interface)
jet will become invalid as the jet thickness approaches the available
layer depth (hmax)/H + 1).
Under these conditions, dynamic pressure
forces will arise which cause flow reversal and which are not included
in the present model formulation.
This condition can only be evaluated
by comparison with experimental observations, as is done in the following.
7.5
Comparison of Analytical and Experimental Results
Figure 7.10 through 7.14 show the predictions and the values
found in experiments 1-5, 26, 30, 31 and 33 (conditions of low F
)
o
for the centerline temperature and the depth to half of the centerline
temperature difference.
The experimental points are from vertical tem-
perature profiles constructed from temperature measurements.
For the
polynomial distributions, the half temperature depth, h 5T, is proportional to the total jet depth, h:
h
5T
= 0.63h
The predictions for experiments 6 and 8 (conditions of high
IF
and high discharge) are also presented in this section although,
o
as stated in Chapter IV, there are some physical observations indicatr
ing that recirculation may have occurred.
123
If recirculation has occurred,
the model formulation of the flow problem becomes invalid, as discussed
earlier.
7.5.1
Low Froude Number Experiments (without Recirculation)
The centerline temperature plots (Figures 7.10 to 7.14, "a"
series) show that the agreement between the predicted and observed
values is very good.
This is consistent with the results of the
sensitivity study performed in Section 7.4.
The analytical model
predicts that the centerline temperature is rather insensitive to
changes in the discharge conditions.
Therefore, slight irregularities
in the circumferential distribution of the discharged flow in the
plexiglas model areexpected to have only a minor effect on the radially
averaged centerline temperature.
The agreement between the theoretical predictions and measured
values for the half temperature depth, h 5T, are satisfactory, as seen
in Figures 7.10 to 7.14, "b" series.
The sensitivity analysis performed
in Section 7.4 for the jet depth shows that it changes almost linearly
with a change in the discharge parameters, provided the jet is confined
by the layer.
Therefore, if the flow is not uniformly distributed,
there will be an angular variation in the depth of the jet.
A large
variation could bias the radially averaged value since the experimental
data is taken at discrete points in the external field.
A detailed
comparison of the vertical temperature structure of the radial jet as
predicted by the model and observed in the experiment is given in Fig.
124
_
1.0
I
1
0.8
I
2
I
3
i
4
I
5
I
r(m)
6
* Run # 1
-4
o
.
Run # 26
# 30
n # 31
0.6
AT
AT
0
0.4
.
0.2
Figure 7.10a
I
I
I
I
I
I
100
200
300
400
500
600
I
I
700
I
800
r/ho
900
Temperature vs. Distance, Experimental Comparison
to Model
1
2
~I-
20
3
I
I
5
4
I
6
I
r(m)
I
* Run # 1
15
h sT
o
Run # 26
+
Run # 30
x
Run # 31
10
h. T(cm)
h
O
10
S
5
*
S
5
100
Figure 7.10b
200
300
400
500
600
700
800
900
r/h
Half-Temperature Depth vs. Distance, Experimental
Comparison to Model
125
1.0
r(m)
0.8
0.6
AT
0
0.4
0.2
100
Figure 7.11a
200
300
400
500
600
700
800
r/h
900
0
Temperature vs. Distance, Experimental Comparison
to Model
,rm
1
2
3
4
I
I
I
I
5
6
I
I
r(m)
Run # 2
10
15
h.5T(
hST
h
0
10
5
4000*
0
·_
Of
·
1
100
Figure 7.11b
200
·
·
·
,
1 1
1
300
400
1 I
500
.
.
1
1
1
600
700
800
5
AI
900
r/h
Half-Temperature Depth vs. Distance, Experimental
Comparison to Model
126
m
c)
1.0
r(m)
0.8
0.6
AT
AT
o
0.4
0.2
100 200 300
Figure 7.12a
zu
.!
400 500
600
700
800
900
r/h
Temperature vs. Distance, Experimental Comparison
to Model
1
I
_r
2
I
3
4
I
!
5
I
6
I
r(m)
10
Run # 3
15
h
h 5T(cm)
. ST
h
o
10
5
5
I
100
Figure 7.12b
I
200
I
I
I
I
300
400
500
600
I
700
800
900
r/h
Half-Temperature Depth vs. Distance, Experimental
Comparison to Model
127
1.0
I
2
I
1
I
3
I
4
I
5
0.8
I
6
r(m)
Run # 4
0.6
AT
AT
o
0.4
0.2
100
100
Figure 7.13a
200
300
200
300
400
I
II
500
600
I900
700
800
900
r/h
o
Temperature vs. Distance, Experimental Comparison
to Model
20
1
2
3
4
5
6
I
I
I
I
I
I
r(m)
Run # 4
10
15
h .5T
h
0o
h T(cm)
_
10
5
5
I . I1
100
Figure 7.13b
I
IiII
00200
300
400
500
600
700
800
900
200
300
400
500
600
700
800
900
Half-Temperature Depth vs. Distance, Experimental
Comparison to Model
128
1.CIj I
I
1
I
2
I
3
r(m)
0.8I
* Run # 5
0.6 I
+ Run # 33
AT
AT
O
0.4
0.2
*
'_
-----100
Figure 7.14a
200
300
I
400
I
500
600
700
800
900
Temperature vs. Distance, Experimental Comparison
to Model
1
2
30
3
r(m)
- 1mj
25
h
20
h
15
5
.ST
h
o
10
5
100
Figure 7.14b
200
300
400
500
600
700
800
900
Half-Temperature Depth vs. Distance, Experimental
Comparison to Model
129
(cm)
7.15 for Exp.*33
The angular variability in the experiment is also
.
indicated.
The error in interpreting the experimental data is 1 to 2
discharge slot heights, h , because scans were taken at discrete levels.
The fit of a temperature profile to the data is usually imperfect; in
which case there is a variance in the estimate of h
.5T'
Furthermore,
as the radial distance from the model approaches the length scale of
the basin, there is not likely to be any time period within which a
steady state approximation is valid, so the data will be biased.
Bear-
ing these considerations in mind, the agreement between the analytical
model and the experiments on h 5T is good; particularly, closer to the
discharge where the major mixing occurs.
In addition to the individual comparison of results for each
experiment, other methods for establishing the goodness of fit between
the analytical model and physical data were developed.
Figure 7.16a
is a graph of the centerline temperature (0.5 cm depth) at a fixed
radial distance of
number, Fr
h
= 148.0 against discharge densimetric Froude
o
, for all the experiments with a layer depth of H/ho =
O
53.5 and a plant geometric ratio of r/h°
= 18.0.
The solid line
represents the predictions based on the analytical model and the points
are the experimental values.
Figure 7.16b is the corresponding graph
for the half temperature depth.
agreement is good.
For both temperature and depth, the
The model does not exhibit any trend for either
130
4)
0
4,
t
.
ChU
>
L
-- -. ,,
+
C
Lu
rrr
Lu
C
L
0
~
L
,4.
E
c(
4,
(U
'n
·
C
Ž'E~
*04)E
0.
V
o
!o
E
·
H0
c
4i
!I
N
-Q)
·
c
a)
I
.:
..
n~
.13
0)
37
a)
4-1~
·`
· I·
·?·
·
·---
C
---- fII-
r 11
!b
4.
-U
C4
0
4
(V)
d
A
4r
.CM
d
4.
C
0
in
II
U
.1)
0
tI3
to
4)
1
4.
uL
W
4)
+
~
.4..~~~~~~~~~~~~~~~....
.4.
IE
.
0
I-W
Vr
*d
C
C)
·.4
(U
H,
m
*
C)
r4 E
Q)
(
.1d'-4
d
'0
U:.----·
· ·.
T
·h
E
U
'· ' · ' ' '.
. . . . . . . .
.'
.r
I
h\\
\
~~~~~~~\
~o\\\ --\\\\~\
~-
·-~ · ·
\\
·
\7\Y-
1~~~~~~~~~~~~~~~~~J
C%
131
o
.a
rz4'-
0.4
0.3
AT
0.2
0.1
0.0
ro
18.0
h
H
ho
Figure 7.16a
=
30
20
10
r
h
40
IFr
o
= 148
53.5
Normalized Temperature Difference vs. Discharge
Densimetric Froude Number at a Radial Distance
r
= 148
o
132
_
It
_
_I_ Ilr_·I__Us__l_l__ell__l
1__1111_1___1__1____-I_
1___^
20
15
hh.5T
h0
10
5
0
10
20
30
40
IFr
o
r
o
h0
h
r-
=
-18.0
= 148
0
= 53.5
0
Figure 7.16b
Normalized Half-Temperature Depth
vs,
Discharge
Densimetric Froude Number at a Radial Distance r
h
= 148
O
133
underprediction or overprediction.
Figures 7.17a and 7.17b are similar
r
=
graphs for those experiments with different geometric properties (
O
36.0 andH /h
= 107) than in the previous set.
Only points for runs 5
and 33 are included since the other experiments with these parameter
The limited agreement shown in these
values appeared to recirculate.
graphs may be due to the experimental inaccuracies discussed earlier.
In general, as the discharge Froude number increases, the
flow approaches that of a pure momentum jet in which buoyant damping
becomes less important and jet spreading becomes sharper.
Figure 7.16a
indicates that the centerline temperature approaches a constant value
for very high discharge Froude numbers.
However, at increasing values
of IFr , the effects of the confined layer also tend to become stronger
o
which can ultimately lead to a breakdown of the jet.
High Froude Number Experiments (with Recirculation)
7.5.2
Figures 7.18a and 7.18b present the comparison of predicted
and measured surface temperatures and half-temperature depths for run 6.
Figures 7.19a and 7.19b are the corresponding predictions for run 8.
Physical observations indicate that recirculation may have occurred in
both of these experiments.
The analytical model also fails to give a
stable jet depth.
r
For the geometric parameters in these experiments, h
134
_
__
__
_
·_
I
_I___
__
_____
I
______
= 36.0
and H/h
o
= 107.0,
numerical instabilities interrupt the integral
marching procedure of the analytical model when the total jet depth
reaches
70% of the total layer depth.
Here return flow velocities
have a greater magnitude than jet velocities.
This is an unstable
situation because jet entrainment is significant and the jet depth is
increasing.
The deeper the jet becomes, the greater is the return
flow velocity.
The jet velocity relative to the return flow velocity
may actually increase (as the model steps out from the discharge
point) causing even greater jet depths and entrainment.
Also in this
situation, the dynamic pressure effects that the return flow exerts
on the jet should not be ignored if the formulation is to be accurate.
A conservative estimate of the onset of recirculation would
be when the jet depth is approximately half the layer depth.
Thus, the
following condition can be assumed to hold for the applicability of the
radial jet model with return flow:
h
max<
0.5
(7.23)
If inequality (7.23) is not satisfied, near field recirculation can be
assumed to occur.
However, the model is not able to predict the degree
of recirculation.
This situation is indicated in Figs. 7.20, which is
a revision of Fig. 7.9, combining the experimental observations with
the theoretical base.
In view of this analysis. run 6 is probably a
borderline case of recirculation caused by angular variations of the
jet depth arising from non-uniformities of discharge distribution at
135
0.20
0.15-
AT
S.,
0.10-
AT
0.05 -
0.00
I
10
I
I
20
30
I
40
I
I
50
60
IFr
r
h
= 36.0
H
ho
Figure 7.17a
h
o
296.0
o
= 107.0
Normalized Temperature Difference vs. Discharge Densimetric Froude Number at a Radial Distance of r
= 296.0
o
136
40
30
h
.5T
h0
20 -
10-
0 -.
1 I
I
10
20
30
r
o
ho
H
H
h
Figure 7.17b
I
I
I
40
50
60
r
hF0
= 36.0
IFr
0
= 296.0
= 107.0
o
Normalized Half-Temperature Depth vs. Discharge Densi= 296.0
metric Froude Number at a Radial Distance of
0O
137
I
I
I
1
2
3
1.0
I
I
4
5
0.8
r(m)
6
Run # 6
0.6
AT
AT
0
0.4
0.2
I1
I
1
200
Figure 7.18a
I
·
I.
II
.
400
600
800
·
1
a
II
I
II
Ia
C
I
I
I
r/ho
1000 1200 1400 1600 1800
Temperature vs. Distance, Experimental Comparison
to Model
1
3
2
5
4
80
6
Ru I
.
6
r(m)
.
25
70I
Run # 6
20
60I
h .5T
h
o
50
15
h 5T(cm)
40
*
.
,
I
I
I0 t
I
0
I
10
3(
I
20
J
10
i
I
I
Figure 7.18b
L
.
200
400
A
.
800
600
I
II_
a
I
I
Half-Temperature Depth vs. Distance, Experimental
Comparison to Model
138
_
I
1000 1200 1400 1600 1800
_·
I
5
1.0
I
I
I
I
I
I-
1
2
3
4
5
6
r(m)
Run # 8
0.8
0.6
AT
AT
O
A
0.4
0.2
I I, _ , l I . 0.
200
Figure 7.19a
400
600
800
1000 1200 1400 1600 1800
r/ho
Temperature vs. Distance, Experimental Comparison
to Model
1
2
3
4
5
WffB
6
r(m)
25
20
h .5T
h
15
h 5T(cm)
O
10
5
200
Figure 7.19b
400
600
800
1000 1200 1400 1600 1800
r/h
0
Half-Temperature Depth vs. Distance, Model
Prediction Only
139.
_________
·
1.u
4n
r
v0
0.8
0.6
hmax
H
0.4
0.2
0.
no
16.
f)
0
20
40
60
80
100
120
140
Fro
Figure 7.20 Maximum Vertical Penetration vs.Quarter
Module Froude Number; (Experimental Surface Jet
Data Points are Included).
140
I__U__··_YIIYIIIII-a_-LYIIIDIIIIL·
_
the plexiglas model.
Inequality (7.23) will be used in the final
chapter to derive a more general design guideline for the assessment
of recirculation with the mixed discharge design.
141
CHAPTER VIII
SEPARATE JET THEORY WITH STAGNANT CONDITIONS
The analytical buoyant surface jet model of Stolzenbach and
Harleman (1971) was chosen for separate discharge jet simulations.
A
user's manual for this model has also been published (Stolzenbach,
Adams and Harleman, 1972).
This integral jet model predicts near field behavior of
buoyant surface jets discharged into an isothermal body of water of
large depth (Figure 8.1).
Several of the characteristics distinguish-
ing the experimentally observed flow field are not included in the
formulation.
These include:
1. the intake flow zone; 2. the return
flow zone; 3. interactions between the multiple jets and 4. the limited
depth.
Thus, at first, the application of the model to the OTEC case
should be exploratory only to determine by comparison with experiments
whether (and under what conditions) these characteristics severely limit
the model applicability.
If not, then the model can be used as a tool
to evaluate data trends and give detailed three-dimensional predictions.
8.1
Analytical Background
As in the radial case, the jet dominated near field can be
broken up into two zones:
the zone of established flow and the zone
of flow establishment.
142
8.1.1
Zone of Established Flow
This zone is characterized by fully developed sheared profiles
of velocity and temperature (Figure 8.1).
The zone extends out from
the zone of flow establishment to the far field region where the discharge flow no longer has turbulent jet type features.
The analysis approach is similar to that used for the radial
buoyant jet.
The time averaged turbulent momentum, heat conservation,
and continuity equations are reduced to ordinary differential equations
through a series of assumptions and integration over the jets crosssectional area.
These assumptions are typical for integral analysis
of buoyant jets:
1.
Steady flow:
2.
Large Reynolds number:
3.
Boussinesq approximation:
at
= 0
viscous terms negligible
density differences are
only important in pressure terms
4.
Hydrostatic pressure
5.
No jet induced motion at large depths:
ap = ap = 0
ax
ay
as z + -
x <<
«
6.
Boundary layer flow
7.
Small density differences
ax
y and a
ay
az
P
<<
1
Pamb.
The assumption of self-similarity of vertical and lateral profiles of
velocity and temperature allows the cross-sectional integration of the
equations resulting in the equations of Table 8.1.
143
These equations
7
(
Velocity
Temperature
X
lb~m
Figure 8.1
Schematic for Buoyant Surface Jet
144
can be solved numerically using given flow conditions at the beginning
of the zone.
The particular similarity functions chosen for this model are
polynomials as assumed by Abramovich (1963) for sheared buoyant jets.
u = uc f(y/b)
f(z/h)
AT = ATc t(y/b)
t(z/h)
where:
= [1 -
f()
t(5) = [1-
3/2]2
3/2]
Two features are particular to the Stolzenbach-Harleman
surface jet model.
First of all, entrainment velocities are assumed
proportional to the jet centerline velocity.
Separate proportionality
coefficients are assumed for lateral and vertical entrainment.
lateral entrainment coefficient is a constant, 0.0495.
The
The vertical
entrainment coefficient is reduced to a function of local Froude
number
(see Eq. 7.20).
The second feature concerns the lateral velocity, V.
It is
analogous to some sort of lateral spreading assumption often employed
in other formulations.
The local lateral velocity due to buoyant
spreading is assumed proportional to:
gradient
fa )
; 2. the local
1. the local normalized density
longitudinal
I-T -
145
velocity,
u; 3. the lateral
Table 8.1
GOVERNING EQUATIONS AND INTEGRAL QUANTITIES
FOR SURFACE JET MODEL BY STOLZENBACH AND HARLEMAN (1971)
Governing Equations
Conservation of Mass
d
dx
Conservation of
= c a u h + c a u b
1
2 v c
c
x -Momentum
dM
dx
dP
dx
Y-Momentum Spreading Equation
d
-
2
h52
kj)C
bhc4 (F) -2 cT
(
Conservation of Heat Flux (Surface Jet)
dH = -c3 K' AT b
Centerline Temperature Boundary Condition (Interface Jet)
d(ATc)
=
dx
O
Integral Quantities
Volume Flux:
Q = |
udnd
A
Momentum Flux:
Pressure Force:
Temperature Flux:
M= I
P =
TA
H
=
pau2
dnd
a
1
[
gApdp] ddt
f
ATudfnd
146
Table 8.1 (Continued)
Parameters
A
=
cross-sectional area of jet
r
=
y/b
=
z/h
c
profile dependent coefficients
i
K' =
= kinematic surface heat loss coefficient [L/T]
K/pc
P
k
I
=
[db
= lateral spreading rate of a non-buoyant surface jet
[dx INB
typically taken as .22
147
spreading rate in excess of a non-buoyant jet
( a|B)
This assumption
is important in the formulation of the y-momentum spreading equation.
8.1.2
Zone of Flow Establishment
This zone extends from the discharge point out to the zone
of established flow.
The mean velocity and temperature profiles are
characterized by central portions that are unsheared.
The zone is
basically a transition from the approximately uniform velocity of the
discharge port to the free-shear velocity distribution in the jet.
The model of Stolzenbach and Harleman carries out a detailed
analysis of this zone which is necessary in cases where the zone of
flow establishment encompasses a significant portion of the near-field
zone.
The large port sizes proposed for OTEC power plants contribute
to the size and significance of the zone.
The solution approach divides the jet cross-section into 4
regions of sheared, partially sheared, and unsheared velocity profiles
(Figure 8.2).
The system of governing equations then contains four
individual equations of continuity and longitudinal momentum linked
through transfer conditions.
The detailed equations and their develop-
ment can be found in Stolzenbach and Harleman (1971).
8.1.3
Interface Jet
For the prototype discharge jet at the density interface,
the temperature along the centerline is constant.
equation needs changing for this case.
148
Only the heat flux
It was replaced by the boundary
t f
c0
E
4,
z
E
I--
c
T
0
U
w
V)
.c-
C
LI)
0
.4i
4
c
()
us
Et
r,4
ri
'4V?
ILL
3
0
I
14..
0
T
wC4
O
0
N,
.1-4
Ua;
L
4)::I
u
n
4
UO(
-)
0
.J
-(
G)
N
Us
-i{
4J
0I->
w
U')
149
$-I
zc
dAT
dx
condition
= 0.
The same polynomial profile shapes of the surface
jet are used in these simulations as well.
8.2
Solution Method
The governing equations are solved by a fourth order Runge-
Kutta integration technique supplied by the IBM Fortran Scientific
The routine was modified slightly to prevent
Subroutine Package.
physically positive variables from ever taking on negative values.
The program
bound.
chooses a step size to meet an inputted error
The solution is stepped out from the discharge point until a
specified distance is reached or until the low velocity and low local
Froude number flow can no longer be considered to exhibit jet-like
behavior.
The results of the calculations are printed out in dimensionless form with u , AT , and
hb-
being the normalizing velocity,
temperature and length scales.
8.3
Model Results
The analysis of the model and the examination of its sensi-
tivity to the discharge parameters IF
8.3.1
and h /b
have been conducted.
Surface Jet
The surface jet behavior and parameter sensitivity have been
examined thoroughly by Stolzenbach, Adams and Harleman (1972).
8.3 illustrates typical model predicted jet behavior.
150
Figure
The jet mixes
ho
20
40
60
2
4
6
l
80
100
1
0.
h
-
-
1Q
SIDE
b
Fh-b
X = Y/h-.b.
nce
2
TOP VIEW
Figure 8.3
Surface Jet Model Preditions for Vertical and Lateral
Jet Dimensions (Discharqe Conditions Correspond to Experiment #13).
151
down to a maximum depth and then gets shallower as lateral spreading
becomes more important.
Lateral spreading is at first moderate but
then increases strongly as buoyant spreading dominates the x momentum
of the jet.
A transition distance, xt, where the local Froude number
equals one, has been defined
(Eq. 5.1).
be the limit of strict applicability of the model.
This appears to
Buoyant spreading
and far field phenomena rather than jet behavior can be expected to
dominate beyond this distance.
The model continues its calculations
beyond xt, until the predicted centerline velocity is .02 of its
original value.
For the conditions encountered in the laboratory, the
inclusion of surface heat loss had little effect on jet behavior for
distances in the order of x t.
Figures 8.4 and 8.5 illustrate the sensitivity of centerline
depth and temperature to changes in the Froude number, Fo
aspect ratio, h /b .
by [h° b]½
and
and the
The depths and temperatures are nondimensionalized
T , respectively.
cause more mixing and deeper jets.
Higher discharge Froude numbers
However, jet solutions for IF
equals 15.0 are insensitive to changes in the aspect ratio over the
range of .2 to 1.5.
The difference in solution for all aspect ratios
in this range is less than the thickness of the curves drawn in Figures
8.5a and 8.5b.
152
T=
hT
Centerline Timnerature vs. Distance,
FPiCure 8.4a
Varyinq
0
100
IF
O
300
200
400
= 0X
Piqure 8.4b
Jet Depth vs. Distance, Varying IF0
153
T=
100
0
200
Fi(-.ure R.5a Centerline
Varying
-- x
X= b
400
300
Tenperntlre
vs.
istance
h /b
15Fo= 15.
10h= ho
5h/
= 1.5
o
/: h.=
- 0.2*
0
I
100
C)
I
200
300
400
x
ibt
Friure
8.5
Varrir.q ho/bo
Jet Depth vs. Distance,
0 0
The curves for the rane 0-2 ho /h -' 1. 5 lie
within the thickness of the line drawn.
154
8.3.2
Interface Jet (Prototype)
The centerline temperature of the interface jet (as described
in 2.2) does not change, since it entrains equally from both sides of
the density interface.
The jet thickness responds similarly as a func-
tion of distance from the same F
o
as the surface jet except that buoyant
damping of turbulence is much more pronounced.
The interface jet is
thinner, has more pronounced buoyant spreading, and loses its jet properties much sooner (smaller x t ) than the surface jet.
Figure 8.6 and 8.7 illustrate the sensitivity of the jet
thickness prediction to F
and h /b .
Actually, a dimensionless "half"
thickness or the jet penetrating above the density interface is plotted.
High discharge Froude numbers indicate a thicker, better mixing jet.
As was the case in surface jets, there is no sensitivity (less than
graphs line thickness) to h /b
over the range of 0.2 to 1.5.
The information of Fig. 8.4b and 8.6 is further summarized
by plotting in Fig. 8.8 the maximum predicted jet thickness, h
a function of F
o
range 0.2 to 1.5.
.
, as
The plot is good for any aspect ratio h /b
0
0
in the
Predictions of vertical penetration based on the
surface jet are conservative with respect to prototype conditions
(interface jet).
The important parameter, recirculation, should be
directly related to the discharge jet's vertical penetration or thickness.
8.4
Comparison of Analytical and Experimental Results
Even though the experiments for the separate jet discharge
155
15
10
5
0
100
0
Fisrure
8,6
-_ x
X-
200
300
Interface Jet Depth vs. Distance,
Varvinc IF 0
156
15
10
_
h
5h-
200
100
0
X
Figure
8.7
x
h-o
Interface Jet Depth vs. Distance
Varying ho/bo
0
0
The range of 0.2< ho/bo 1.5 is contained in
the thickness of the curve drawn.
157
300
20.0
rr
_~
~
A
1:.U
~
~
~
~~UFCSURFACE
10.0
hmax
5.0
n0
..
0.
10.
I
I
20.
30.
F*
I
40.
I
50.
0
Figure 8.8
Dimensionless aximum Jet Thickness vs.
Discharge Densimetric Froude Number for Surface Jet
and Interface
Jet.
158
configuration do not allow a direct classification into cases without
and with recirculation, the comparison is carried out in two sections
(analogous to the radial jet case, Section 7.5):
First, experiments
with low Froude number conditions and no recirculation; second, experiments with high Froude number conditions showing some tendency for recirculation (based on the observations in Section 5.3).
8.4.1
Low Froude Number Experiments
The centerline temperature plots (Figures 8.9 to 8.12, "a"
series) show that the agreement between the predicted and observed values
is very good.
Data points appear for each individual jet of the half or
full model experiment.
temperature.
There is a slight tendency to underpredict the
This could be due to the model not accounting for the heat
entrained from the lateral return flow.
The underprediction tendency
is slight, indicating that the importance of this effect is only secondary.
The agreement between theoretical predictions and measured
half temperature depths is satisfactory.
The scatter is partly due to
the turbulent fluctuations in the temperature field.
Experimental steady
state durations did not always allow sufficient data to be taken for
good estimates of mean temperature.
The fit also depends significantly
on the temperature similarity profile assumed in the model.
Figure 8.13
illustrates the correctness of this profile assumption for experiment 37.
159
A
2
4
4 F1%I
I.
V
I
4;
-
5
7
6
I
o Run
0
0.8
'
8I
.
s
r (m)
10
Run 35
+ Run
O.
39
0.6
.,
0A
o
0.4
0.2
F
.
.o
o
o
nA
0
Figure 8.9a
50
+I
.,>
100
150
°
80
200
hb
00
Temperature vs. Distance, Experimental Comparison
to Model
(m)
2.5
2.0
h
,5T
[h-.b
1.5
hI5T
(crm)
1.0
0.5
C
0
Figure 8.9b
50
100
200
/
/0
Half Temperature Depth vs. Distance, Experimental
Comparison to Model
160
r(m)
AT
.J
0
Figure 8.10a
vu
1U
1M0
200
Temperature vs. Distance, Experimental Comparison
to Model
(m.)
h
h
.5T
(m.)
I
/h.
Figure 8.10b
b.
Half Temperature Depth vs. Distance, Experimental
Comparison to Model
161
r (m.)
aT
·To
.
50
Figure 8.11a
100
150
.
200
Temperature vs. Distance, Experimental Comparison
to Model
hLT
flW
[N-b.~i
h.5T
(C m)
Figure 8.11b
Half Temperature Depth vs. Distance, Experimental
Comparison to Model
162
0I
2
1
3
I
1.0 r
5·
4
6·
7·
0 Run
8
r (m.
_ .
v
..
-
.
13
Run 37
+ Run .41
*
0.8
8
0.6
0.4
.0o
,+
0
0.2
*
0.0
0
.
, -8*
*r
200
100
.
I
400
300
_,
l
Figure 8.12a
0
hH.b
Temperature vs. Distance, Experimental Comparison
to Model
1
3
2
5
4
7
6
8
r (ml
-T-
8.0
Run 13
0
* Run 37
+ Run 41
16
6.0
h.5T
12
h5T
4.0
8 (cm.
:
2.0
a
o
0
0o
,%,
IL J % !
0
100
4
o
o
I
I
200
300
I
400
/Nkbhb
Figure 8.12b
Half Temperature Depth vs. Distance, Experimental
Comparison to Model
163
0
L
> 1U
4J
C
4c
LL
,
o
(.
E
ro
+'~c
L
E
'0
q
V
4
4_
"JH
a
E
Dt
.,,
4.
C)r'
. _4
+
En
lz
U4
Ul
.
H
..
d
4a
o
t
·
M
o
U4J
4
c
H
C4 C~~~~~~,,
-
curb
E.
dX~--I ar
..,
164
All experimental values of centerline temperature and half temperature
depth are based on data similar to that displayed in Figure 8.13.
8.4.2
High Froude Number Experiments
The centerline temperature plots (Figures 8.14 to 8.15, "a"
The
series) show good agreement between predicted and observed values.
half temperature depth plots (Fig. 8.14 to 8.15, "b" series) also show
good agreement, though with more experimental scatter.
This is in
contrast to radial experiments for which the maximum jet depth was
not predictable for the high Froude number range.
The important differ-
ence in radial experiments is the absence of a lateral return flow zone.
If a radial jet at some point penetrates more than half of the basin
depth, less than 50% of the cross-sectional area at that radius is
The average return flow velocity exceeds
available for return flow.
that of the discharge flow.*
This is not the case for separate jets.
The lateral return flow zone provides a large cross-sectional area for
return flow even when the jet reaches the basin floor.
The dynamic
pressure effects of a fast underlying return flow are not present.
The analytical jet model considers only an infinitely deep
receiving water body.
Since the discharge jets of experiment 15 did
reach the basin floor, the analytical surface jet predictions will
become less appropriate for higher discharge Froude numbers.
Empirical
By continuity, the magnitude of flow moving away from the model must
be equal to the returning flow.
165
1.0
I
0C
33
2
1
.
4
6I
5
77
°,o Run
8·
r (~
·
·
14
0,
S
AT
&T
0.4
-o
0.2
O-
0.0
0
I
I
I
I
I
I
200
100
.
I
r
400
300
/Hob,O
Figure 8.14 a
C'3
Temperature vs. Distance, Experimental
Comparison to Model
1
2
3
I
I
II
5
4
6
7
I
I
16.0
o,
8
I
'
r (m)
Run 14
32
12.0
h
24
h.b
0
8.0
h.5T
16
o
(cm.)
oo
4.0
·
S
~~o
8
.
/
j
*
*
g
s
|
Io~~~~~~~~~~~~~~~~0
*0
0.0
0
Figure 8.14b
100
200
400
/h
Half Temperature Depth vs. Distance, Experimental
Comparison to Model
166
Y-·------·--l.l-----....-
300
0
3
2
1
1.0
4
5
7
6
Run
o,.
a r (mi
r--9ff
15
0.8
0
0.6
AT
"To
0.4
S
0.2
0.0
I
I.
0
Figure 8.15a
0I
16.0 F
.
.
I.
400
300
200
100
Temperature vs. Distance, Experimental
Comparison to Model
4
3
2
1
I
5
6
7
r (m]
8
rI
.
o,.
Run 15
32
12.0
24
0
0
.5T
8.0
I
16 (cm.)
0
4.0 I
I
·
0
Figure 8.15b
100
·
·
200
·
$
·
300
400
/Ijhj
Half Temperature Depth vs. Distance, Experimental Comparison to Model
167
methods exist to take into account the effect of interaction with the
bottom (Jirka, Abraham and Harleman, 1975).
8.4.3
Discharge Froude Number Sensitivity
In the analytical model, the local densimetric Froude number,
L
is indicative of the local jet behavior.
dependence of vertical entrainment on
This is caused by the
FL (Eq. 7.20).
For example, the
transition from the near field jet zone to the far field zone is signaled
by a
FL
of 1.0.
Maximum jet penetration occurs for
FL
in the range
2.1 to 2.4.
It was desired to compare the model to experimental data at
a location of similar behavior for all the stagnant separate jet experiments.
Therefore, a local Froude number value was chosen.
The ana-
lytical model was used to predict at which location each of the experimental runs would reach this value of
FL .
Figures 8.16 to 8.18 are
the result of comparing experimental jet surface temperature and half
temperature depth at these locations with the values predicted by the
model.
The non-dimensionalized depths and temperatures appear as func-
tions of the discharge Froude number,
IF
.
(As Section 8.3 demonstrated,
there should be little or no dependence on the port aspect ratio,
ho/bo).
The solid line is the analytical model prediction.
These points
are averages of the data taken from each of the jets in the experiment.
168
0.3
L=
0.2
1.0
.35
.10
aT
aTo
0.1
3
',*41
37a 1
~14
37
0.0
0.
20.
10.
Figure 8,16a
40.
30.
50.
Jet Centerline Temperature vs. Discharge
Froude Number at the Position
of IFL =
1.0
9.0
14
6.0
h.5T
/-h..
41
3.0
FL= 1.0
37
1
I
0.0
0.
l
10.
20.
F
40.
30.
3
Figure 8.16b
Jet Penetration Depth vs. Discharge
Froude Number
at the Position
169
of
FL
=
1.0
50.
0,3
FL-= 2.0
\.35
Q2
_
10o 39
AT
aT.
12\
0.1
.41
1;9s
37
141
S
0.0
·
·
30.
20.
10.
0C
1
40.
50.
Fo
Figure
.17a
Jet Centerline Temperature vs. Discharge
Froude Number at the Position of IFL= 2.0
0
9.0
6.0
_
14
h. 5T
oKb
37
3.0
_
12
FL = 2.0
.41
3
X
11
39/
*35
10
0.0
II
0.
I
20.
10.
F*
30.
40.
Figure 8.17b Jet Penetration Depth vs. Discharge
Froude Numberat the Position of FL = 2.0
170
··
__1
_1__·1·__·_1_____1111_11_111
-
--
50.
-- V
'..35
0.3
-
FL =
3.0
12
AT
AT.
41
11
1
0.1
-
i
I
I
I
0.0
I
-
20.
10.
O.
30.
F.*
40.
-
I
50.
Figure 8.18a
Jet Centerline Temperature vs. Discharge
Froude Number
at the Position
of
= 3.0
9.0
-_
F
= 3.0
6.0
h.5T
o-0b.
3.0
-
12
11
!
0.0
0.
10.
I
!
20.
Figure 8.18b
30.
I
40.
Jet Penetration Depth vs. Discharge
Froude Number at the Position of IFL=
3.0 .
171
50.
Experimental data and analytical predictions are in good
agreement in each case.
F
There is a trend in the temperature plot for
= 1.0 [at the transition to the far field].
Measured temperatures
exceed the predicted values indicating reduced dilution probably caused
by the lateral return flow (Section 5.3.2).
Also, there seems to be a
trend to overpredict the penetration depth for 1%
jets are still deepening].
172
= 3.0 [while the
CHAPTER IX
CONCLUSIONS AND DESIGN CONSIDERATIONS
9.1
Summary
Experimental and analytical studies have been carried out to
determine the characteristics of the temperature and velocity fields
induced in the surrounding ocean by the operation of an OTEC plant.
In particular, the condition of recirculation, i.e., the re-entering
of mixed discharge water back into the plant intake,was of interest
due to its adverse effect on plant efficiency.
The studies were
directed at the mixed discharge concept, in which the evaporator and
condenser water flows are exhausted jointly at the approximate level
of the ambient ocean thermocline.
The ocean was assumed to be distinct-
ly stratified and to have a uniform current velocity.
The studies were aimed at a 100 MW "standard" OTEC plant
operating at a thermocline depth of about 70 m (see Section 2.2).
Considerable variations in the size and the design and operating conditions of this plant were considered.
The two major types of discharge
geometries were a radial slot around the periphery of the plant and
four separate ports at a 900 angle with each other.
In order to simplify the procedure, the experimental program
(at an approximate scale of 1:200) was carried out by considering the
173
flow in the upper layer of the ocean only.
The surface layer was
modeled in an inverted manner and the actual OTEC discharge at the
ocean thermocline was simulated by a model discharge at the surface.
The conservativeness of this procedure, regarding the prediction of any
recirculation, was demonstrated by analytical techniques.
The analytical studies consisted of the formulation and
application of integral jet models for both the radial jet case and
the separate jet case.
The models predictions are in good agreement
with the experimental data and can be used for understanding the
discharge behavior (data trends) and for establishing design strategies.
9.2
Conclusions
Based on the combined experimental and analytical results,
the following conclusions are made:
i)
The major characteristic zones of the temperature and
flow field of the OTEC plant are a jet mixing zone and an intake flow
zone in the near field (of the order of a few hundred meters in extent)
and density and ambient current zones in the far-field.
Of these,
the jet discharge mixing zone is probably the most important one.
Effective jet entrainment causes the accumulation of large quantities
of mixed water at the edge of this near-field zone.
ii)
This accumulation of fluid masses, which ultimately
may result in intake recirculation, is counteracted by two mechanisms:
174
buoyant convection in the form of density currents and convection by
ambient ocean current.
critical one:
stagnant ocean.
iii)
The role of the first seems to be the more
the recirculation potential seems highest for the
Small currents (
0.10 m/s) approach this condition.*
The jet mixing is predominantly controlled by the
discharge velocity (ie.by the discharge area for a given flow rate).
The shape of the discharge opening ports appears to have a secondary
effect (with radial jets and separate jets representing extremes in
the shape effect).
This is reflected in the definition of the govern-
ing discharge parameter, a densimetric Froude number
,*-
F
o
u
=
0
where u
o
= discharge velocity
.Apn
v = relative density difference between discharge and upper
-
O
ocean layer (i.e., one half of the total relative density
difference between the upper and lower layer for the
mixed discharge concept).
g
= gravitational acceleration
The experimental program was limited to this current range. The
conclusion is based in part on the expected reduced recirculation
potential for strong ocean currents due to its adequate convection
mechanism.
175
A
= characteristic area
o
= area
of discharge =
of one of the four discharge
ports
or area
of a
quarter section of radial discharge slot.
iv)
The strength of the intake flow is given by another
densimetric Froude number,
Qi
F.
Po
H5
-gH
where Q=
H
(9.2)
intake flow (equal to evaporator flow)
= upper layer depth
v)
A condition of recirculation is generally associated
with a high degree of jet discharge mixing and high intake flow rate,
i.e., a combination of large values for F
and IF
.
(Alternatively,
this situation is described by another set of dependent parameters,
namely
Fr
and H/h
as has been done in the analytical model for
o
the radial
jet).
vi)
It appears that a "standard" 100 MW OTEC plant under
baseline stratification conditions can be designed to operate without
near-field recirculation by using a low velocity (low F
discharge mode at the ocean thermocline.
) mixed
In fact, this recirculation-
free operating condition appears to be possible up to the 200 MW range.
In each case, the area is taken as the "half area" above the thermocline for the symmetric discharge geometry.
176
9.3
Design Considerations
At this point
in time,
it appears
that the primary
design
objective for the interaction of an OTEC plant with the ocean environment must be the prevention of a degree of recirculation that would
cause a "significant" loss of heat engine efficiency.
loss carries a considerable economic burden.
An efficiency
Based on an analysis of
early OTEC engineering designs Lavi (1975) proposed the following
relationship between plant unit costs C and the thermal resource ATo.
C ~ ATo
As
an example,
-2.5
(9.3)
if the base parameter
is AT
0
= 200 C, a steady
tion which results in a thermal resource depression by 1C
recircula-
(5%) would
require an investment increase of 13% per unit power produced.
Thus,
even modest recirculation influences can have a significant economic
impact.
Other OTEC design objectives concerning environmental impacts and plant costs are not as well defined at this time.
example,
For
possibly desirable nutrient enrichment would be caused by
the lodging of the condenser discharge somewhere in the photosynthetic
mixed layer.
But this strategy may be at odds to possibly detrimental
effects of thermocline changes and drops in the mixed layer temperature.
Plant construction costs implied by different sized ports
and levels of discharge are not obvious.
The economy of scale for
constructing larger capacity (e.g., 200-300 MW plants) plants is not
177
known.
Deeper discharge levels would seem to mean greater costs due
to the need of placing the heat exchangers and heat engine machinery
further from the ocean surface.
Larger ports imply more piping
(greater material costs) and slower discharge velocities (lower pumpAs in the case of environmental impacts, the importance
ing costs).
of these combinations (in designing an optimal OTEC power plant) have
not been quantified.
The desirable strategy of exactly where in the stratified
ocean, the OTEC discharge waters should have their sink is not clear.
The answer is largely dependent on the analysis of the intermediate
field behavior (of order 10 km), its associated phases of buoyant
spreading and mass transport and possible interaction with adjacent
plants and the far-field behavior (of order 100 km to basin scale)
with its effects on thermocline dynamics, surface heat transfer and
basin-wide circulations.
Thus, the following design considerations are restricted to
the primary objective, namely control of near-field recirculation.
However, the experimental and analytical results of this report can be
used for the assessment of other design issues as well.
9.3.1
Mixed versus Non-Mixed Discharges
The most basic design choice involves the relative orienta-
tion of the condenser and the evaporator discharge flows.
mode" concept, the
In the "mixed
flows are either combined before being discharged
close enough together so that the jets merge near the outlet.
178
*I
__
_
__
I
In the
"unmixed mode", the discharge flows are separated such that there is
little or no interaction between them.
It is of utmost importance to differentiate between the
consequences of recirculation (usually given as a percentage value)
for these two discharge modes.
The factor of interest is the temper-
ature decrease 6T of the available thermal gradient AT (see Eq. 9.3).
Simple mass and heat balances for the two design cases yield the
6T
following relationships between the normalized resource decrease AT
and the recirculation fraction, r, of discharged water back into the
evaporator intake:
Mixed Discharge:
aT = r {
where
AT Q + (AT -AT )Q
Q (1 r ) + Q
}
(9.4a)
Qe = evaporator flow rate
Qc = condenser flow rate
AT
e
AT
C
= temperature drop of evaporator flow
= temperature drop of condenser flow.
For the usual case of Qe ' Q
e
T
XV
c
and AT
e
buAT
X
c
_r
AT
AT
2-r
(9.4b)
2-r
Unmixed Discharge (only evaporator flow recirculating):
AT
=
AT
A-Y
r
(-V )
AT
e
AT
(9.4c)
179
Example:
The disparate consequences of discharge recirculation for the
two cases is demonstrated
for the case,
AT = 200 C, AT
e
= 20 C
andr = 25%.
For the mixed discharge,
T
AT
14
.3%
or
T = 2.86°C
For the unmixed discharge,
3.3%
T =
or
T = 0.66°C
The considerably stronger sensitivity of the mixed discharge
mode to agiven value of recirculation is contrasted by its much lower
likelihood of occurence.
In fact with proper design, the mixed mode
can be made to have zero recirculation, while recirculation seems much
more
likely in the unmixed mode.
The experimental results can be used
as evidence for this conclusion:
The major part of the experimental program was devoted to
the mixed mode of discharge.
a
X
In this mode, the discharge waters have
110 C temperature differential from the evaporator intake water.
It
is the buoyancy effects of this temperature differential that counteracts recirculation tendencies.
The basic result of the experimental
program was that near-field recirculation could be wholly prevented for
this mode discharging into a stagnant or slow current environment.
There would be no loss of plant efficiency for plant sizes up to 200 MW.
180
I_____
_
The "unmixed mode" discharge, on the other hand, has a
greater tendency for evaporator intake recirculation than the "mixed
mode."
This is because of lower temperature and density differentials
(2 or 30 C rather than 11 C) between evaporator intake and discharge
water.
Three of the experiments conducted with smaller temperature
differences (Exp. 8, 54 and 55, see Chapter 4) showed definite recirculation based on dye measurements and visual observations.
The observed
temperature rise, however, was small (less than 0.03°F) and would not
suggest a complete disqualification of the "unmixed" discharge mode
from further consideration.
The data, however, suggest that in the
OTEC range of up to 200 MW the "mixed" mode can be designed with a much
higher confidence regarding the prevention of recirculation than the
"unmixed" mode.
9.3.2
Design Formula
In Chapter 7, the analytical model for discharge mixing was
used in conjunction with the experimental results to determine a criteria for the onset of recirculation
h
max
H
> 0.5
%
(9.5)
In essence, the formula states that recirculation of mixed discharge
water back into the evaporator intake will occur if the jet zone, hmax,
occupies more than 50% of the available mixed layer depth, H.
This can
be interpreted as a "blocking" of the ambient water flow toward the
intake.
Since both the analytical studies (Chapter 8) and the experi-
181
mental data (Chapters 4 and 5) have demonstrated that similar discharge
mixing occurs regardless of jet discharge geometry (radial or separate),
Eq. 7.23, can be applied in either case as long as hma is given as
*
*
a function of a Froude number F
. As noted earlier F
is a Froude
0
0
number which only depends on the jet discharge area but is independent
of its geometry.
Using analytical predictions of jet thickness and assuming
simple dependences on the parameters
, r/h
Fr
and H/ho, Eq. 9.5 is
o
modified to
I
F
o
5/2
h
r
8.7
+ 6.3II(-)
5
+ 2.6 x 10
] < 1
(9.6)
o
applicable for the radial jet geometry and experimental parameter
ranges (Table 2.5).
With the definition of the discharge Froude number
*, Eg. 9.1, the intake Froude number, Eq. 9.2, and the identity
F
ho = (2/b)(Ao/ro) a more general formula is derived:
r
r
4
-2
1 + 4.0 (-_)
]1.4
F.
F
+ 4.4 x 10
5/2
(o-2)
0
]
< 1
(9.7)
0
applicable for any symmetric discharge geometry.
Eq. 9.6
can be modi-
fied by using the actual design parameters
3
u
r
1/2
Ap
qi
where
u
r
2
Q.
u
Qi
]
M g00
= discharge velocity
Qi = evaporator flow rate
182
-
-I
_~
urH
5/2
+ 2.6 x 10
3
] < 1
(9.8)
r
- plant radius at discharge ports
H = upper layer depth
Ap
o
- =
relative density difference of a mixed discharge with respect
P
to the upper ocean layer
g = gravitational acceleration.
The set of operating parameters above is complete in the
sense of determining the recirculation potential of an OTEC plant with
a radial, mixed, discharge jet.
Recirculation sensitivities can then
be studied from Eq. 9.8 by varying one of the parameters while leaving
the others fixed:
Recirculation potential
increases
Recirculation potential
decreases
Increased discharge
velocity,
f"u
°"
Increased flow rate
(i.e., plant size)
"Qi
Increased mixed
layer depth,
"H"
Increased thermal
gradient,
"Aop"
P
Secondary Recirculation
potential changes
Plant radius,
"r
The range of the other design parameters determine the sensitivity of
this parameter.
Thus Eq. 9.8 can be used to assess the recirculation likelih.ood for specific design concepts and choices of design parameters and
183
to estimate the effect of variable conditions (such as the seasonal
changes in Ap/p
and H).
The restriction to the mixed discharge concept
has to be kept in mind, however.
The following examples illustrate the use of the design
formula:
A)
Standard 100 MW Plant (see Table 2.2):
rO = 23 m, Qi
H
3
500
/5, u
= 2 m/s, APo/p = 0.003 (11°C),
50 to 100 m.
The left hand side of Eq. 9.8 is denoted by C.
C = 0.4 to 0.1
C <<1
B)
(as H = 50 to 100 m)
No recirculation likely.
200 MW Plant:
Qi = 1000 m3/s, u
= 8 m/s (comparable to discharge velocities
for submerged diffusers), r
C = 0.9 to 0.5
C < 1
(as H
= 23 m
50 to 100 m)
No recirculation likely, although critical value would
be approached for small H and decreases in AT .
C)
400 MW Plant:
3
Qi = 2000 m 3/s,
u
= 8 m/s , r
= 23 m
C = 15
to 0.4
C
Incipient recirculation likely.
X
1
(as H = 50 to 100 m)
On the basis of such perliminary calculations it might be concluded that
a maximum possible OTEC plant size which carries a degree of conservativeness (e.g., for the effect of weak currents) would be on the order
of 200 MW.
Although note that Eq. 9.8 does not predict the degree,r,of
184
_____
recirculation but only whether recirculation takes place or not.
9.3.3
Intake Design
The sensitivity of te
external flow field to evaporator
intake design was not treated in the experimental program or the
analytical models.
The optimal intake level would appear to be as
close to the surface as possible (above discharge ports) without causing excessive flow velocities and surface drawdown.
Due to its potential flow characteristics, the flow field is
expected to be nearly independent of the intake opening dimensions and
horizontal location except in the immediate vicinity of the intake.
The intake induced flow accounts for only a portion of the double sink
return flow.
Specific intake design parameters will only effect that
port-ionand only near the intake opening.
The portion of return flow eventually reaching the plant intake is small.
If at a certain distance, a discharge jet has reached
dilution of 10, then only 10% of the return flow at that point will
eventually enter the intake.
Considering the stable dilutions encount-
ered in experiments (Fig. 4.7, 5.6 and 6.4), the intake induced return
flow is overwhelmed by that induced by jet entrainment.
For low Froude number discharge configurations, dilution is
small and the intake induced return flow is more important.
Single sink
studies which account for intake geometry and variable stratification
(such as Katavola, 1975) may be useful in predicting intake design
sensitivities.
185
REFERENCES
1.
Abramovich, G.N., The Theory of Turbulent Jets, M.I.T. Press,
M.I.T., Cambridge, Massachusetts, (1963)
2.
Albertson, M.L., Dai, Y.B., Jensen, R.A., Rouse, H., "Diffusion
of Submerged Jets", Trans. ASCE, 115 (1950)
3.
Bathen, K.H., Kamins, R.M., Kornreich, D., Mocur, J.E.T., "An
Evaluation of Oceanographic and Socio-Economic Aspects of a
Nearshore Ocean Thermal Energy Conversion Pilot Plant in Subtropical Waters", NSF-RANN Grant No. AER74-17421 A01, (1975)
4.
Carnahan, B., Luther, H.A., Wilkes, J.O., Applied Numerical
Methods, John Wiley and Sons, Inc., New York (1969)
5.
Craya, A., "Recherches Theoriques sur L'Ecoulent de Couches
Superposees de Fluides de Densites Differntes", LaHouille
Blanche, Jan.-Feb. (1949)
6.
Daily, J.W. and Harleman, D.R.F., Fluid Dynamics, AddisonWesley Publishing Co., Reading, Massachusetts (1973)
7.
Dugger, G.L. (ed.), "Proceedings, Third Workshop on Ocean Thermal
Energy Conversion (OTEC)", Houston, Texas, May 8-10, 1975, Johns
Hopkins University, Applied Physics Laboratory, SR75-2 (1975)
8.
Fuglister, F.C., Atlantic Ocean Atlas, Temperature and Salinity
Profiles and Data from the International Geophysical Yeat of
1957-1958, The Woods Hole Oceanographic Atlas Series, Vol. 1,
Woods Hole, Massachusetts (1960)
9.
Jirka, G.H., Abraham, G., and Harleman, D.R.F., "An Assessment
of Techniques for Hydrothermal Prediction", R.M. Parsons Laboratory for Water Resources and Hydrodynamics, Department of Civil
Engineering, M.I.T., Technical Report No. 203 (1975)
10.
Katavola, D.S., "An Experimental Study of Three-Dimensional
Selective Withdrawal from a Thermally Stratified Fluid System",
S.M. Thesis, Department of Ocean Engineering, M.I.T. (1975)
11.
Koh, R.C.Y, and Fan, L.N., Mathematical Models for the Prediction
of Temperature Distributions Resulting from the Discharge of
Heated Water into Large Bodies of Water, EPA Water Pollution
Control Research Series 16130 DWO (1970)
186
_
I
___
12.
Lavi, A., "Final Report: Solar Sea Power Project," CarnegieMellon University, January 1975.
13.
Lee, J.H.W., Jirka, G.H. and Harleman, D.R.F., "Stability and
Mixing of a Vertical Round Buoyant Jet in Shallow Water", R.M.
Parsons Laboratory for Water Resources and Hydrodynamics, Department of Civil Engineering, M.I.T., Technical Report No. 195
(1974)
14.
Schlicting, H., Boundary Layer Theory, McGraw-Hill Book Co.,
(1968)
15.
Slotta, L.S. and Charbeneau, R.J., "Ambient Current Effects on
Vertical Selective Withdrawal in a Two-Layer System", Final
Project Report, OWR Proj. No. A-025-ORE, Office of Water Research
and Technology, U.S. Department of the Interior (1975)
16.
Stolzenbach, K.D. and Harleman, D.R.F., "An Analytical and Experimental Investigation of Surface Discharges of Heated Water",
R.M. Parsons Laboratory for Water Resources and Hydrodynamics,
Department of Civil Engineering, M.I.T., Technical Report No.
135 (1971)
187
APPENDIX A
EXPERIMENTAL ISOTHERM
DATA
Isotherms labeled in percent of discharge temperature
difference: (AT,/AT
0 )*100
188
APPENDIX A. 1
RADIAL STAGNANT EXPERIMENTS
(see Tables 4.1 & 4.2 for discharge parameter values)
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4J
a)
C)
4.
a)
r_
LCr)
* H
H
sp
a1)
a)
r)
lw
0
i
C)
-W
co
4-4i
a)
U
=r
a4
r)
k4
0)
4!J
r-rt
0c
40c
IL
*i-4
p
H
-H
Co
P4
C14
Cm
C)
U)
.lq
ri-4
C)
a)
cn
H
X
Lr
CD
ci
-4
C)
0
-
H
ad
o
U
v-
250
-- ·- ·
--·-----9·I1IBW
ilP--1
I·----.-ur*--l--------n---ul--rr*
4-
P.
4
C
O
U
0
I
a)
N
ra
a)
Ln
St
x
o4
,
"i
a)
0)
a)
C
a)
F/
ao
4-J
r-
O
O
D~
a)
.H
00
co
u
0,
.
m
u
a)
E
:5
til.
.11
Lo-
4,
0-
u
251
'4
0
_o
APPENDIX
C
FAST PROBE TEMPERATURE MEASUREMENTS
iAT = mean value of temperature
record (above ambient)
c = standard deviation of ten-
perature
252
+.2_
nT 1.4 3C - +
=
= 0.07 oC
1.5
~" aT=160C --
+
4l T
0
-.2
CT = 0.141
-*40·;1---.c
D
E
P 3.(
4T= 1,10C-+
T
cr= 0.20°C
H
(c
AT 0,85C
ur- 0.19c
J
T = 0.32 C
- = 0.23°C
+.f
-Zc
0
+.4
7.5
11
= 0.32°C
r - 0. 15° C
- Vo
- 40
Temperature Variation with
00
Temp. Profile
15
Seconds
Time
30
F - 14.3
Figure C. 1
Experiment
@ r =
55.
H/ho =55.
60
0.91m
253
45
r/ho = 18.
T= 1.30C--- +
---------
o-= 0.03°C
D
E
P
T
A T = 1.05C -+
H
=0.09c /
i
;-·;.
(Cr
,
AT = 0.55 C
- = 0.05C
;;··
U ,T = O° C
Temp.Profi le
Figure C.2
J
I-IIII
I
0.
~ ~ ~~
'
~
~
-
TP~b
LI /~;~+h
r~ll
n~·r+t:--~
cILLUI
IlIUII
Wl III
Irlt
C
Vdl-
I II
II
15.
Seconds
45.
30.
Experiment #61
@ r=
2.13m
Fr =14.4
254
H/ho =55.
r/h
o
=18.
- T
1.06°C.
TT
-
+
r- = .13° C.
4aT = 0.80
l
0-
.14
°
C.+
C
D
E
P
I
I
H
q AT=0.4 3°C
U-
= .16°C
(c n
'1
9
a4 T = 0.09
C.
,
- =:0.11° C.
1°
+
I
11.
-
.
_
Temp. Profile
Temper ature Variat ion w'ith Time
!.
f-
0.
Figure C.3
Experiment #51
@ r= 0.91m
255
I
I!
15.
I
m
I
Seconds
45.
30.
F
=79.5
r
110.
H/h
°
I O,
60.
rVh
=36.
-
___
_
_
___
f
I
_
OE.
.
--
=0.48 C -+
U- . 02°C
3.C
J T = 0.4 3C --+
---
cr- 0.05tc
+.1
+
I
--EIT=0.37C-+
,n
6.C
°
05D
4
cr = 0.05 C
i
,~1
AT -
ff[
iII
-. 05
Vyi 1
i
1:
-1 o1
D
E
P
T
a,T
9.C
H
=
0.28°C +
- = 0.04°C
(cm.)
11.9
-+
UaT=0.2
3C
C = 0.04°C
I+
14.9
,UI T = 0.13C
0- = 0.0o9C
18.3
+
li AT= 0.05C
= 0,08 0C
Temp Profile
Temperature
I
I
0
Figure
C. 4
Experiment
@ r = 2.13m
Variation with Time
I
I
45
5Seconds
#58
F
80.7
256
H/ h = 110.
60
r,/h o = 36.
APPENDIX D
DEVELOPMENT OF THE EQUATIONS FOR THE RADIAL JET DISCHARGE
The analysis of the steady-state external flow and temperature
fields generated by a radially discharging OTEC power plant is best
conducted in cylindrical coordinates as angular variations do not exist.
By examining only the near field the discharge flow can be treated as
a boundary layer (jet) using the turbulent fluid transport equations
(Daily and Harleman, 1973).
With these assumptions and using the
hydrostatic approximation, the appropriate conservation equations for
mass, momentum and heat are:
Continuity
1
r
aw
+
aru
ar
0
3z
(D.1)
Momentum
P
z:
0
+ auw
aru
1
r:
ap
ar
r
=
P +pg
aTrz
P
(D2)
(D.3)
az
Heat Transport
apc AT
u
arrP
apc AT
+
w
azP
=
257
(D.4)
r,z
coordinate
=
directions;
origin
taken
at the center
of the
plant; z is positive downward in the direction of gravity
u,w
=
velocities in r and z directions, respectively
p
=
pressure
=
turbulent shear stress
=
specific heat per unit volume
=
temperature deviation from reference temperature
T
rz
pc
P
AT
Integrating over the vertical extent of the jet; from the upper
boundary, bl, to the lower boundary, b2 ; and applying Leibnitz rule
the equations become:
Continuity
b2
r
d
dr
b
db
db
1
+ WIbl
udz]-ulb2 dz
[r
b2 ~z +ulb
0
=0
(D.5)
b
Momentum
r:
b2
[rf u2 dz]-u2 Ib
d
p
db2
+ u2
1
b
+ (u,w) l1 }
b
1
[dr d]+rzbl
-
b2
b
z:
db
(D.6)
b
b2
2
2
P'b
=
1
J
pgdz
(D.7)
bl
258
Heat Transport
d
r
d
dr [r
bdb
'
uATdz]-(uAT)b
2
dr
+
(wAT)
1
(AT)l
dr
2
b
b
(D.8)
The radial momentum equation simplifies further, since, by
definition, there is no turbulent shear at the edge of the jet,
Trz b
Ib
=rz
=.
These equations are general enough to apply to both the model and
the prototype discharge fields.
It is in considering the boundary
conditions that the two situations vary.
D.1
Equations for the Jet at the Free Surface (Model)
Figure B.1 is a diagram of the vertical velocity and temperature
distribution in the fully established (i.e. self-similar) jet region
for the surface discharge situation in the model.
the jet centerline, b 1 = 0.
b2 = ho.
The water surface is
The lower boundary of the jet is the depth,
The velocity in the region below the jet,
b, constitutes
the return flow velocity for the intake and entrained flow make-up water.
Thus, the velocity within the jet is defined as:
u
=
(D.9)
ur+b
where ur is the relative velocity with respect to the return flow.
259
Urc
Figure D.1 Schematic Diagram of Surface Jet Vertical
Velocity and Temperature Profiles.
260
_
_____
I_
(i)
Substituting for u gives the general set of equations for
an arbitrary return flow velocity ub.
Continuity
h
r
dr
[
(ur-ub)dz
-ulh dr +h
=
(D.lO)
Momentum
h
0
h
f
dz
Heat Transport
1
d
d
r dr
[rJ (Ur-ub)ATdz]
r
=
0
(D.12)
o
(ii)
For the case of the confined layer where the volumetric flow
is equal in opposite directions, as found in the model situation, the
return flow velocity is determined:
h
u dz
0
The
=
-ubH
(D.13)
conservation equations become:
The conservation equations become:
261
Continuity
h
-d h
(1 - )
(1
d
[r
udz]
=
(D.14)
-Wl h
Momentum
h
Id
r
dr
[r
h
2
Hr2
u dz] -
[r(
d
h
1
Urdz)2
]
H2
o
dh
1
I
P
J
0
o
dr
H r
UrdZ)]
(rUrdz)
( u dz)2
2
dr [rh(J
(D.15)
H
dp dz
dr
0
Heat Transport
Id d r
r
dr
h
[
u ATdz ]
Ur
[
1
Hr
h
d
o
(iii)
[r(
h
udz)(
o
ATdz)]
= 0
(D.16)
o
In the case of a semi-infinite region with no intake to
cause a return flow the equations are simply:
Continuity
h
r
dr
[r
udz]
=
-wh
(D.17)
0
262
LI_-____-_ __1II____II.~~-------IO-~~~~~___·I_
IIIIU
)
I--m~~~-·-I_~~-_I _I^II~~~~_~_____~~)_I____
___I___^
~~P~I
Momentum
h
1r
ddr
1u
h
ddz] I=
2
dr
p
(D.18)
o
o
Heat Transport
h
d
[ri uTdz]
=
0
(D.l9)
o
For small return flows, ub approaches zero, the equations for
case (ii) approach those for case (iii).
D.2
Equations for the Jet at the Thermocline (Prototype)
The physical situation in the prototype discharge jet is differ-
ent.
tions.
Figure B.2 is a diagram of the velocity and temperature distribuThe receiving water upper layer is considerably smaller than the
lower layer.
thermocline.
Thus, the flow must be asymmetric with respect to the
However, as a first approximation, for small return flows
in the upper layer, the velocity distribution in the upper half of the
jet can be assumed symmetric with the velocity distribution in the lower
half.
The temperature distribution, though, is antisymmetric, providing
a smooth transition between the upper layer temperature, T 1, and the
lower layer temperature, T2.
The interface temperature is constant
and equal to the plant discharge temperature, To = (T1 +T 2 )/2.
263
r
'Uk
z
0
Figure D.2 Schematic Diagram of Interface Jet Vertical
Velocity and Temperature Profiles.
264
?
___
_IQ
IICI-L----·..II*_
-I
-I
_I_._
_I_._.^_
_._._._
The continuity and momentum equations are the same in form for
the submerged jet as for the surface discharge case, equations (B.14)
The heat transport equation differs from the surface jet
and (B.15).
because the heat flux in the jet is changing due to entrainment of water
with different temperatures in the upper and lower layers.
The heat conservation equation is:
iddr .[r
r
h
uATdz]
dr
=
w
h (T 1 - T 2
)
(D.20)
-h
where AT is the temperature difference from T2.
Substituting for u, the equation becomes:
h
r
Id
r
u ATdz]
r dr
-1
d
h
h
u dz)(j ATdz)]
o
o
=
WI-h(T1-T2 )
265
_
.I...-I--
---
(D.21)
APPENDIX E
DEVELOPMENT OF THE PRESSURE DERIVATIVE IN THE
VERTICALLY INTEGRATED MOMENTUM EQUATION
The pressure derivative in the vertically integrated radial momentum equation can be reduced to a product of the unknown variables
and known coefficients.
Pressure can be divided into two parts:
=
P
ph=
d
Ph+Pd
(E.1)
hydrostatic pressure
=
dynamic pressure
Density can also be divided into two parts:
P
=
Pa-AP
(E.2)
Ap
=
[AT
(E.3)
p
=
ambient (reference) density
8
=
thermal expansion coefficient
a
Using the hydrostatic pressure approximation in the treatment of
the jet equations (Appendix D) the dynamic pressure component is
neglected in the following, pd = 0.
The hydrostatic pressure is
further defined as:
266
I
l""~~c~~-
-;I-P--."~""."Y111···L-----·--11·1
· I·----_III·-·L^-U·IU··lll.
I^-I---·-_m_--
· ·
II
Ill^·C
P·--l·ll**.
z
(p-Ap)gdz
Ph=
(E.4)
(z positive down)
o
The pressure derivative can be written:
dz
d
dr
dr
dr
dr
[- Apgdz]
LjpsZJ
0
(E.5)
Outside the jet region, z = , no motion is assumed, so that
dp
=
0
=-
dr
(E.6)
at z =
dr
[
gdz]
(E.7)
Subtracting Equation (C.7) from Equation (C.5) gives
dr
dr
LJpz
[J
(E.8)
pgdz ]
So
-fdp
dz =
dr
o
Apgdz'd
- r
-d
(E.9)
z
The limit 'h' is taken since the values outside the jet are zero.
267
-----------..-__
E.1
Integrated Pressure Term for the Surface Jet
Knowing the temperature distribution function, g(z/h), where h
is the jet depth, the pressure term for the surface discharge case
becomes:
h
I
dr dz
=
ddr
(OgtTh2 G)
2)
(E.10)
o
where G2 is an integration constant
h
E.2
h
G2
=
f
AT
=
ATcg(z/h)
g(z?/h)dzdz
(E.ll)
(see Figure D.1)
Integrated Pressure Term for the Submerged Jet at the Interface
Due to the antisymmetric temperature distribution the discharge
jet at the thermocline is negatively buoyant with respect to the upper
layer and positively buoyant with respect to the lower layer.
Limiting
the analysis to jets with width, h, smaller than the thermocline depth,
H
H << 1, it is possible to approximate the integrated pressure term
H,
by:
-
dp dz = -h
dr [
Apdz'dz] +
h z
[|
o
Apdz'dz] = -2
[gATh
G2 ]
z
(E.12)
where AT
(see Figure D.2)
c~~~seFgr
.2
=
ATcg(z/h).
268
I--
--
This result implies that the upper and lower layer buoyancy forces
are equal and the same as for a surface jet with the same temperature
distribution.
269